An elliptic curve is given by a Weierstrass model

y^2+a_1xy+a_3y = x^3+a_2x^2+a_4x+a_6,

whose discriminant is non-zero. Affine points on `E`

are represented as
two-component vectors `[x,y]`

; the point at infinity, i.e. the identity
element of the group law, is represented by the one-component vector
`[0]`

.

Given a vector of coefficients [a_1,a_2,a_3,a_4,a_6], the function
`ellinit`

initializes and returns an *ell* structure. (An additional
optional argument allows to specify the base field in case it cannot be
inferred from the curve coefficients.) This structure contains data needed by
elliptic curve related functions, and is generally passed as a first argument.
Expensive data are skipped on initialization: they will be dynamically
computed when (and if) needed, and then inserted in the structure. The
precise layout of the *ell* structure is left undefined and should never
be used directly. The following member functions are available,
depending on the underlying domain.

Returns the value at s = 1 of the derivative of order r of the L-function of the elliptic curve e.

? e = ellinit("11a1"); \\ order of vanishing is 0 ? ellL1(e) %2 = 0.2538418608559106843377589233 ? e = ellinit("389a1"); \\ order of vanishing is 2 ? ellL1(e) %4 = -5.384067311837218089235032414 E-29 ? ellL1(e, 1) %5 = 0 ? ellL1(e, 2) %6 = 1.518633000576853540460385214

The main use of this function, after computing at *low* accuracy the
order of vanishing using `ellanalyticrank`

, is to compute the
leading term at *high* accuracy to check (or use) the Birch and
Swinnerton-Dyer conjecture:

? \p18 realprecision = 18 significant digits ? e = ellinit("5077a1"); ellanalyticrank(e) time = 8 ms. %1 = [3, 10.3910994007158041] ? \p200 realprecision = 202 significant digits (200 digits displayed) ? ellL1(e, 3) time = 104 ms. %3 = 10.3910994007158041387518505103609170697263563756570092797 [...]

The library syntax is `GEN `

.**ellL1_bitprec**(GEN e, long r, long bitprec)

Sum of the points z1 and z2 on the elliptic curve corresponding to E.

The library syntax is `GEN `

.**elladd**(GEN E, GEN z1, GEN z2)

Computes the coefficient a_n of the L-function of the elliptic curve
E/ℚ, i.e. coefficients of a newform of weight 2 by the modularity theorem
(Taniyama-Shimura-Weil conjecture). E must be an `ell`

structure
over ℚ as output by `ellinit`

. E must be given by an integral model,
not necessarily minimal, although a minimal model will make the function
faster.

? E = ellinit([0,1]); ? ellak(E, 10) %2 = 0 ? e = ellinit([5^4,5^6]); \\ not minimal at 5 ? ellak(e, 5) \\ wasteful but works %3 = -3 ? E = ellminimalmodel(e); \\ now minimal ? ellak(E, 5) %5 = -3

If the model is not minimal at a number of bad primes, then the function will be slower on those n divisible by the bad primes. The speed should be comparable for other n:

? for(i=1,10^6, ellak(E,5)) time = 820 ms. ? for(i=1,10^6, ellak(e,5)) \\ 5 is bad, markedly slower time = 1,249 ms. ? for(i=1,10^5,ellak(E,5*i)) time = 977 ms. ? for(i=1,10^5,ellak(e,5*i)) \\ still slower but not so much on average time = 1,008 ms.

The library syntax is `GEN `

.**akell**(GEN E, GEN n)

Computes the vector of the first n Fourier coefficients a_k corresponding to the elliptic curve E defined over a number field. If E is defined over ℚ, the curve may be given by an arbitrary model, not necessarily minimal, although a minimal model will make the function faster. Over a more general number field, the model must be locally minimal at all primes above 2 and 3.

The library syntax is `GEN `

.
Also available is **ellan**(GEN E, long n)`GEN `

, which
returns a **ellanQ_zv**(GEN e, long n)`t_VECSMALL`

instead of a `t_VEC`

, saving on memory.

Returns the order of vanishing at s = 1 of the L-function of the
elliptic curve e and the value of the first non-zero derivative. To
determine this order, it is assumed that any value less than `eps`

is
zero. If no value of `eps`

is given, a value of half the current
precision is used.

? e = ellinit("11a1"); \\ rank 0 ? ellanalyticrank(e) %2 = [0, 0.2538418608559106843377589233] ? e = ellinit("37a1"); \\ rank 1 ? ellanalyticrank(e) %4 = [1, 0.3059997738340523018204836835] ? e = ellinit("389a1"); \\ rank 2 ? ellanalyticrank(e) %6 = [2, 1.518633000576853540460385214] ? e = ellinit("5077a1"); \\ rank 3 ? ellanalyticrank(e) %8 = [3, 10.39109940071580413875185035]

The library syntax is `GEN `

.**ellanalyticrank_bitprec**(GEN e, GEN eps = NULL, long bitprec)

Let E be an `ell`

structure as output by `ellinit`

, defined over
a number field or a finite field 𝔽_q. The argument p is best left
omitted if the curve is defined over a finite field, and must be a prime
number or a maximal ideal otherwise. This function computes the trace of
Frobenius t for the elliptic curve E, defined by the equation #E(𝔽_q)
= q+1 - t (for primes of good reduction).

When the characteristic of the finite field is large, the availability of
the `seadata`

package will speed the computation.

If the curve is defined over ℚ, p must be explicitly given and the function computes the trace of the reduction over 𝔽_p. The trace of Frobenius is also the a_p coefficient in the curve L-series L(E,s) = ∑_n a_n n^{-s}, whence the function name. The equation must be integral at p but need not be minimal at p; of course, a minimal model will be more efficient.

? E = ellinit([0,1]); \\ y^2 = x^3 + 0.x + 1, defined over Q ? ellap(E, 7) \\ 7 necessary here %2 = -4 \\ #E(F_7) = 7+1-(-4) = 12 ? ellcard(E, 7) %3 = 12 \\ OK ? E = ellinit([0,1], 11); \\ defined over F_11 ? ellap(E) \\ no need to repeat 11 %4 = 0 ? ellap(E, 11) \\ ... but it also works %5 = 0 ? ellgroup(E, 13) \\ ouch, inconsistent input! *** at top-level: ellap(E,13) *** ^----------- *** ellap: inconsistent moduli in Rg_to_Fp: 11 13 ? Fq = ffgen(ffinit(11,3), 'a); \\ defines F_q := F_{11^3} ? E = ellinit([a+1,a], Fq); \\ y^2 = x^3 + (a+1)x + a, defined over F_q ? ellap(E) %8 = -3

If the curve is defined over a more general number field than ℚ,
the maximal ideal p must be explicitly given in `idealprimedec`

format. If p is above 2 or 3, the function currently assumes (without
checking) that the given model is locally minimal at p. There is no
restriction at other primes.

? K = nfinit(a^2+1); E = ellinit([1+a,0,1,0,0], K); ? fa = idealfactor(K, E.disc) %2 = [ [5, [-2, 1]~, 1, 1, [2, -1; 1, 2]] 1] [[13, [5, 1]~, 1, 1, [-5, -1; 1, -5]] 2] ? ellap(E, fa[1,1]) %3 = -1 \\ non-split multiplicative reduction ? ellap(E, fa[2,1]) %4 = 1 \\ split multiplicative reduction ? P17 = idealprimedec(K,17)[1]; ? ellap(E, P17) %6 = 6 \\ good reduction ? E2 = ellchangecurve(E, [17,0,0,0]); ? ellap(E2, P17) %8 = 6 \\ same, starting from a non-miminal model ? P3 = idealprimedec(K,3)[1]; ? E3 = ellchangecurve(E, [3,0,0,0]); ? ellap(E, P3) \\ OK: E is minimal at P3 %11 = -2 ? ellap(E3, P3) \\ junk: E3 is not minimal at P3 | 3 %12 = 0

**Algorithms used.** If E/𝔽_q has CM by a principal imaginary
quadratic order we use a fast explicit formula (involving essentially
Kronecker symbols and Cornacchia's algorithm), in O(log q)^2.
Otherwise, we use Shanks-Mestre's baby-step/giant-step method, which runs in
time Õ(q^{1/4}) using Õ(q^{1/4}) storage, hence becomes
unreasonable when q has about 30 digits. Above this range, the `SEA`

algorithm becomes available, heuristically in Õ(log q)^4, and
primes of the order of 200 digits become feasible. In small
characteristic we use Mestre's (p = 2), Kohel's (p = 3,5,7,13), Satoh-Harley
(all in Õ(p^{2} n^2)) or Kedlaya's (in Õ(p n^3))
algorithms.

The library syntax is `GEN `

.**ellap**(GEN E, GEN p = NULL)

Deprecated alias for `ellheight(E,P,Q)`

.

The library syntax is `GEN `

.**bilhell**(GEN E, GEN z1, GEN z2, long prec)

Let E be an `ell`

structure as output by `ellinit`

, defined over
ℚ or a finite field 𝔽_q. The argument p is best left omitted if the
curve is defined over a finite field, and must be a prime number otherwise.
This function computes the order of the group E(𝔽_q) (as would be
computed by `ellgroup`

).

When the characteristic of the finite field is large, the availability of
the `seadata`

package will speed the computation.

If the curve is defined over ℚ, p must be explicitly given and the function computes the cardinality of the reduction over 𝔽_p; the equation need not be minimal at p, but a minimal model will be more efficient. The reduction is allowed to be singular, and we return the order of the group of non-singular points in this case.

The library syntax is `GEN `

.
Also available is **ellcard**(GEN E, GEN p = NULL)`GEN `

where p is not
**ellcard**(GEN E, GEN p)`NULL`

.

Changes the data for the elliptic curve E
by changing the coordinates using the vector `v = [u,r,s,t]`

, i.e. if x'
and y' are the new coordinates, then x = u^2x'+r, y = u^3y'+su^2x'+t.
E must be an `ell`

structure as output by `ellinit`

. The special
case v = 1 is also used instead of [1,0,0,0] to denote the
trivial coordinate change.

The library syntax is `GEN `

.**ellchangecurve**(GEN E, GEN v)

Changes the coordinates of the point or
vector of points x using the vector `v = [u,r,s,t]`

, i.e. if x' and
y' are the new coordinates, then x = u^2x'+r, y = u^3y'+su^2x'+t (see also
`ellchangecurve`

).

? E0 = ellinit([1,1]); P0 = [0,1]; v = [1,2,3,4]; ? E = ellchangecurve(E0, v); ? P = ellchangepoint(P0,v) %3 = [-2, 3] ? ellisoncurve(E, P) %4 = 1 ? ellchangepointinv(P,v) %5 = [0, 1]

The library syntax is `GEN `

.
The reciprocal function **ellchangepoint**(GEN x, GEN v)`GEN `

inverts the coordinate change.**ellchangepointinv**(GEN x, GEN ch)

Changes the coordinates of the point or vector of points x using
the inverse of the isomorphism attached to `v = [u,r,s,t]`

,
i.e. if x' and y' are the old coordinates, then x = u^2x'+r,
y = u^3y'+su^2x'+t (inverse of `ellchangepoint`

).

? E0 = ellinit([1,1]); P0 = [0,1]; v = [1,2,3,4]; ? E = ellchangecurve(E0, v); ? P = ellchangepoint(P0,v) %3 = [-2, 3] ? ellisoncurve(E, P) %4 = 1 ? ellchangepointinv(P,v) %5 = [0, 1] \\ we get back P0

The library syntax is `GEN `

.**ellchangepointinv**(GEN x, GEN v)

Converts an elliptic curve name, as found in the `elldata`

database,
from a string to a triplet [*conductor*, *isogeny class*,
*index*]. It will also convert a triplet back to a curve name.
Examples:

? ellconvertname("123b1") %1 = [123, 1, 1] ? ellconvertname(%) %2 = "123b1"

The library syntax is `GEN `

.**ellconvertname**(GEN name)

n-division polynomial f_n for the curve E in the variable v. In standard notation, for any affine point P = (X,Y) on the curve, we have [n]P = (φ_n(P)ψ_n(P) : ω_n(P) : ψ_n(P)^3) for some polynomials φ_n,ω_n,ψ_n in ℤ[a_1,a_2,a_3,a_4,a_6][X,Y]. We have f_n(X) = ψ_n(X) for n odd, and f_n(X) = ψ_n(X,Y) (2Y + a_1X+a_3) for n even. We have f_1 = 1, f_2 = 4X^3 + b_2X^2 + 2b_4 X + b_6, f_3 = 3 X^4 + b_2 X^3 + 3b_4 X^2 + 3 b_6 X + b8, f_4 = f_2(2X^6 + b_2 X^5 + 5b_4 X^4 + 10 b_6 X^3 + 10 b_8 X^2 + (b_2b_8-b_4b_6)X + (b_8b_4 - b_6^2)),... For n ≥ 2, the roots of f_n are the X-coordinates of points in E[n].

The library syntax is `GEN `

where **elldivpol**(GEN E, long n, long v = -1)`v`

is a variable number.

k being an even positive integer, computes the numerical value of the
Eisenstein series of weight k at the lattice w, as given by
`ellperiods`

, namely

(2i π/ω_2)^k (1 + 2/ζ(1-k) ∑_{n ≥ 1} n^{k-1}q^n / (1-q^n)),

where q = exp(2iπ τ) and τ := ω_1/ω_2 belongs to the
complex upper half-plane. It is also possible to directly input w =
[ω_1,ω_2], or an elliptic curve E as given by `ellinit`

.

? w = ellperiods([1,I]); ? elleisnum(w, 4) %2 = 2268.8726415508062275167367584190557607 ? elleisnum(w, 6) %3 = -3.977978632282564763 E-33 ? E = ellinit([1, 0]); ? elleisnum(E, 4, 1) %5 = -47.999999999999999999999999999999999998

When *flag* is non-zero and k = 4 or 6, returns the elliptic invariants g_2
or g_3, such that
y^2 = 4x^3 - g_2 x - g_3
is a Weierstrass equation for E.

The library syntax is `GEN `

.**elleisnum**(GEN w, long k, long flag, long prec)

Returns the quasi-periods [η_1,η_2]
attached to the lattice basis *w* = [ω_1, ω_2].
Alternatively, *w* can be an elliptic curve E as output by
`ellinit`

, in which case, the quasi periods attached to the period
lattice basis `E.omega`

(namely, `E.eta`

) are returned.

? elleta([1, I]) %1 = [3.141592653589793238462643383, 9.424777960769379715387930149*I]

The library syntax is `GEN `

.**elleta**(GEN w, long prec)

Let ω := dx / (2y+a_1x+a_3) be the invariant differential form
attached to the model E of some elliptic curve (`ellinit`

form),
and η := x(t)ω. Return n terms (`seriesprecision`

by default)
of f(t),g(t) two power series in the formal parameter t = -x/y such that
ω = f(t) dt, η = g(t) dt:
f(t) = 1+a_1 t + (a_1^2 + a_2) t^2 +...,
g(t) = t^{-2} +...

? E = ellinit([-1,1/4]); [f,g] = ellformaldifferential(E,7,'t); ? f %2 = 1 - 2*t^4 + 3/4*t^6 + O(t^7) ? g %3 = t^-2 - t^2 + 1/2*t^4 + O(t^5)

The library syntax is `GEN `

where **ellformaldifferential**(GEN E, long precdl, long n = -1)`n`

is a variable number.

The elliptic formal exponential `Exp`

attached to E is the
isomorphism from the formal additive law to the formal group of E. It is
normalized so as to be the inverse of the elliptic logarithm (see
`ellformallog`

): `Exp`

o L = Id. Return n terms of this
power series:

? E=ellinit([-1,1/4]); Exp = ellformalexp(E,10,'z) %1 = z + 2/5*z^5 - 3/28*z^7 + 2/15*z^9 + O(z^11) ? L = ellformallog(E,10,'t); ? subst(Exp,z,L) %3 = t + O(t^11)

The library syntax is `GEN `

where **ellformalexp**(GEN E, long precdl, long n = -1)`n`

is a variable number.

The formal elliptic logarithm is a series L in t K[[t]] such that d L = ω = dx / (2y + a_1x + a_3), the canonical invariant differential attached to the model E. It gives an isomorphism from the formal group of E to the additive formal group.

? E = ellinit([-1,1/4]); L = ellformallog(E, 9, 't) %1 = t - 2/5*t^5 + 3/28*t^7 + 2/3*t^9 + O(t^10) ? [f,g] = ellformaldifferential(E,8,'t); ? L' - f %3 = O(t^8)

The library syntax is `GEN `

where **ellformallog**(GEN E, long precdl, long n = -1)`n`

is a variable number.

If E is an elliptic curve, return the coordinates x(t), y(t) in the
formal group of the elliptic curve E in the formal parameter t = -x/y
at oo :
x = t^{-2} -a_1 t^{-1} - a_2 - a_3 t +...
y = - t^{-3} -a_1 t^{-2} - a_2t^{-1} -a_3 +...
Return n terms (`seriesprecision`

by default) of these two power
series, whose coefficients are in ℤ[a_1,a_2,a_3,a_4,a_6].

? E = ellinit([0,0,1,-1,0]); [x,y] = ellformalpoint(E,8,'t); ? x %2 = t^-2 - t + t^2 - t^4 + 2*t^5 + O(t^6) ? y %3 = -t^-3 + 1 - t + t^3 - 2*t^4 + O(t^5) ? E = ellinit([0,1/2]); ellformalpoint(E,7) %4 = [x^-2 - 1/2*x^4 + O(x^5), -x^-3 + 1/2*x^3 + O(x^4)]

The library syntax is `GEN `

where **ellformalpoint**(GEN E, long precdl, long n = -1)`n`

is a variable number.

Return the formal power series w attached to the elliptic curve E,
in the variable t:
w(t) = t^3 + a_1 t^4 + (a_2 + a_1^2) t^5 +...+ O(t^{n+3}),
which is the formal expansion of -1/y in the formal parameter t := -x/y
at oo (take n = `seriesprecision`

if n is omitted). The
coefficients of w belong to ℤ[a_1,a_2,a_3,a_4,a_6].

? E=ellinit([3,2,-4,-2,5]); ellformalw(E, 5, 't) %1 = t^3 + 3*t^4 + 11*t^5 + 35*t^6 + 101*t^7 + O(t^8)

The library syntax is `GEN `

where **ellformalw**(GEN E, long precdl, long n = -1)`n`

is a variable number.

Given a genus 1 plane curve, defined by the affine equation f(x,y) = 0,
return the coefficients [a_1,a_2,a_3,a_4,a_6] of a Weierstrass equation
for its Jacobian. This allows to recover a Weierstrass model for an elliptic
curve given by a general plane cubic or by a binary quartic or biquadratic
model. The function implements the f ` ⟼ `

f^* formulae of Artin, Tate
and Villegas (Advances in Math. 198 (2005), pp. 366--382).

In the example below, the function is used to convert between twisted Edwards coordinates and Weierstrass coordinates.

? e = ellfromeqn(a*x^2+y^2 - (1+d*x^2*y^2)) %1 = [0, -a - d, 0, -4*d*a, 4*d*a^2 + 4*d^2*a] ? E = ellinit(ellfromeqn(y^2-x^2 - 1 +(121665/121666*x^2*y^2)),2^255-19); ? isprime(ellcard(E) / 8) %3 = 1

The elliptic curve attached to the sum of two cubes is given by

? ellfromeqn(x^3+y^3 - a) %1 = [0, 0, -9*a, 0, -27*a^2]

**Congruent number problem:.**
Let n be an integer, if a^2+b^2 = c^2 and a b = 2 n,
then by substituting b by 2 n/a in the first equation,
we get ((a^2+(2 n/a)^2)-c^2) a^2 = 0.
We set x = a, y = a c.

? En = ellfromeqn((x^2 + (2*n/x)^2 - (y/x)^2)*x^2) %1 = [0, 0, 0, -16*n^2, 0]

For example 23 is congruent since the curve has a point of infinite order, namely:

? ellheegner( ellinit(subst(En, n, 23)) ) %2 = [168100/289, 68053440/4913]

The library syntax is `GEN `

.**ellfromeqn**(GEN P)

Returns the coefficients [a_1,a_2,a_3,a_4,a_6] of a fixed elliptic curve with j-invariant j.

The library syntax is `GEN `

.**ellfromj**(GEN j)

If E is an elliptic curve over the rationals, return a ℤ-basis of the
free part of the Mordell-Weil group attached to E. This relies on
the `elldata`

database being installed and referencing the curve, and so
is only available for curves over ℤ of small conductors.
If E is an elliptic curve over a finite field 𝔽_q as output by
`ellinit`

, return a minimal set of generators for the group E(𝔽_q).

The library syntax is `GEN `

.**ellgenerators**(GEN E)

Let E be an `ell`

structure as output by `ellinit`

attached
to an elliptic curve defined over a number field. This function calculates
the arithmetic conductor and the global Tamagawa number c.
The result [N,v,c,F,L] is slightly different if E is defined
over ℚ (domain D = 1 in `ellinit`

) or over a number field
(domain D is a number field structure, including `nfinit(x)`

representing ℚ !):

***** N is the arithmetic conductor of the curve,

***** v is an obsolete field, left in place for backward compatibility.
If E is defined over ℚ, v gives the coordinate change for E to the
standard minimal integral model (`ellminimalmodel`

provides it in a
cheaper way); if E is defined over another number field, v gives a
coordinate change to an integral model (`ellintegral`

model provides it
in a cheaper way).

***** c is the product of the local Tamagawa numbers c_p, a quantity
which enters in the Birch and Swinnerton-Dyer conjecture,

***** F is the factorization of N,

***** L is a vector, whose i-th entry contains the local data
at the i-th prime ideal divisor of N, i.e.
`L[i] = elllocalred(E,F[i,1])`

. If E is defined over ℚ, the local
coordinate change has been deleted and replaced by a 0; if E is defined
over another number field the local coordinate change to a local minimal
model is given relative to the integral model afforded by v (so either
start from an integral model so that v be trivial, or apply v first).

The library syntax is `GEN `

.**ellglobalred**(GEN E)

Let E be an `ell`

structure as output by `ellinit`

, defined over
ℚ or a finite field 𝔽_q. The argument p is best left omitted if the
curve is defined over a finite field, and must be a prime number otherwise.
This function computes the structure of the group E(𝔽_q) ~ ℤ/d_1ℤ
x ℤ/d_2ℤ, with d_2 | d_1.

If the curve is defined over ℚ, p must be explicitly given and the function computes the structure of the reduction over 𝔽_p; the equation need not be minimal at p, but a minimal model will be more efficient. The reduction is allowed to be singular, and we return the structure of the (cyclic) group of non-singular points in this case.

If the flag is 0 (default), return [d_1] or [d_1, d_2], if d_2 > 1.
If the flag is 1, return a triple [h,*cyc*,*gen*], where
h is the curve cardinality, *cyc* gives the group structure as a
product of cyclic groups (as per *flag* = 0). More precisely, if d_2 > 1,
the output is [d_1d_2, [d_1,d_2],[P,Q]] where P is
of order d_1 and [P,Q] generates the curve.
**Caution.** It is not guaranteed that Q has order d_2, which in
the worst case requires an expensive discrete log computation. Only that
`ellweilpairing(E, P, Q, d1)`

has order d_2.

? E = ellinit([0,1]); \\ y^2 = x^3 + 0.x + 1, defined over Q ? ellgroup(E, 7) %2 = [6, 2] \\ Z/6 x Z/2, non-cyclic ? E = ellinit([0,1] * Mod(1,11)); \\ defined over F_11 ? ellgroup(E) \\ no need to repeat 11 %4 = [12] ? ellgroup(E, 11) \\ ... but it also works %5 = [12] ? ellgroup(E, 13) \\ ouch, inconsistent input! *** at top-level: ellgroup(E,13) *** ^-------------- *** ellgroup: inconsistent moduli in Rg_to_Fp: 11 13 ? ellgroup(E, 7, 1) %6 = [12, [6, 2], [[Mod(2, 7), Mod(4, 7)], [Mod(4, 7), Mod(4, 7)]]]

If E is defined over ℚ, we allow singular reduction and in this case we return the structure of the group of non-singular points, satisfying #E_{ns}(𝔽_p) = p - a_p.

? E = ellinit([0,5]); ? ellgroup(E, 5, 1) %2 = [5, [5], [[Mod(4, 5), Mod(2, 5)]]] ? ellap(E, 5) %3 = 0 \\ additive reduction at 5 ? E = ellinit([0,-1,0,35,0]); ? ellgroup(E, 5, 1) %5 = [4, [4], [[Mod(2, 5), Mod(2, 5)]]] ? ellap(E, 5) %6 = 1 \\ split multiplicative reduction at 5 ? ellgroup(E, 7, 1) %7 = [8, [8], [[Mod(3, 7), Mod(5, 7)]]] ? ellap(E, 7) %8 = -1 \\ non-split multiplicative reduction at 7

The library syntax is `GEN `

.
Also available is **ellgroup0**(GEN E, GEN p = NULL, long flag)`GEN `

, corresponding
to **ellgroup**(GEN E, GEN p)*flag* = 0.

Let E be an elliptic curve over the rationals, assumed to be of (analytic) rank 1. This returns a non-torsion rational point on the curve, whose canonical height is equal to the product of the elliptic regulator by the analytic Sha.

This uses the Heegner point method, described in Cohen GTM 239; the complexity is proportional to the product of the square root of the conductor and the height of the point (thus, it is preferable to apply it to strong Weil curves).

? E = ellinit([-157^2,0]); ? u = ellheegner(E); print(u[1], "\n", u[2]) 69648970982596494254458225/166136231668185267540804 538962435089604615078004307258785218335/67716816556077455999228495435742408 ? ellheegner(ellinit([0,1])) \\ E has rank 0 ! *** at top-level: ellheegner(E=ellinit *** ^-------------------- *** ellheegner: The curve has even analytic rank.

The library syntax is `GEN `

.**ellheegner**(GEN E)

Global Néron-Tate height h(P) of the point P on the elliptic curve
E/ℚ, using the normalization in Cremona's *Algorithms for modular
elliptic curves*. E must be an `ell`

as output by `ellinit`

; it
needs not be given by a minimal model although the computation will be faster
if it is.

If the argument Q is present, computes the value of the bilinear form (h(P+Q)-h(P-Q)) / 4.

The library syntax is `GEN `

.
Also available is **ellheight0**(GEN E, GEN P, GEN Q = NULL, long prec)`GEN `

(Q omitted).**ellheight**(GEN E, GEN P, long prec)

x being a vector of points, this
function outputs the Gram matrix of x with respect to the Néron-Tate
height, in other words, the (i,j) component of the matrix is equal to
`ellbil(E,x[i],x[j])`

. The rank of this matrix, at least in some
approximate sense, gives the rank of the set of points, and if x is a
basis of the Mordell-Weil group of E, its determinant is equal to
the regulator of E. Note our height normalization follows Cremona's
*Algorithms for modular elliptic curves*: this matrix should be divided
by 2 to be in accordance with, e.g., Silverman's normalizations.

The library syntax is `GEN `

.**ellheightmatrix**(GEN E, GEN x, long prec)

Look up the elliptic curve E, defined by an arbitrary model over ℚ,
in the `elldata`

database.
Return `[[N, M, G], C]`

where N is the curve name in Cremona's
elliptic curve database, M is the minimal model, G is a ℤ-basis of
the free part of the Mordell-Weil group E(ℚ) and C is the
change of coordinates change, suitable for `ellchangecurve`

.

The library syntax is `GEN `

.**ellidentify**(GEN E)

Initialize an `ell`

structure, attached to the elliptic curve E.
E is either

***** a 5-component vector [a_1,a_2,a_3,a_4,a_6] defining the elliptic
curve with Weierstrass equation
Y^2 + a_1 XY + a_3 Y = X^3 + a_2 X^2 + a_4 X + a_6,

***** a 2-component vector [a_4,a_6] defining the elliptic
curve with short Weierstrass equation
Y^2 = X^3 + a_4 X + a_6,

***** a character string in Cremona's notation, e.g. `"11a1"`

, in which
case the curve is retrieved from the `elldata`

database if available.

The optional argument D describes the domain over which the curve is defined:

***** the `t_INT`

1 (default): the field of rational numbers ℚ.

***** a `t_INT`

p, where p is a prime number: the prime finite field
𝔽_p.

***** an `t_INTMOD`

`Mod(a, p)`

, where p is a prime number: the
prime finite field 𝔽_p.

***** a `t_FFELT`

, as returned by `ffgen`

: the corresponding finite
field 𝔽_q.

***** a `t_PADIC`

, O(p^n): the field ℚ_p, where p-adic quantities
will be computed to a relative accuracy of n digits. We advise to input a
model defined over ℚ for such curves. In any case, if you input an
approximate model with `t_PADIC`

coefficients, it will be replaced by a lift
to ℚ (an exact model "close" to the one that was input) and all quantities
will then be computed in terms of this lifted model, at the given accuracy.

***** a `t_REAL`

x: the field ℂ of complex numbers, where floating
point quantities are by default computed to a relative accuracy of
`precision`

(x). If no such argument is given, the value of
`realprecision`

at the time `ellinit`

is called will be used.

***** a number field K, given by a `nf`

or `bnf`

structure; a
`bnf`

is required for `ellminimalmodel`

.

***** a prime ideal 𝔭, given by a `prid`

structure; valid if
x is a curve defined over a number field K and the equation is integral
and minimal at 𝔭.

This argument D is indicative: the curve coefficients are checked for
compatibility, possibly changing D; for instance if D = 1 and
an `t_INTMOD`

is found. If inconsistencies are detected, an error is
raised:

? ellinit([1 + O(5), 1], O(7)); *** at top-level: ellinit([1+O(5),1],O *** ^-------------------- *** ellinit: inconsistent moduli in ellinit: 7 != 5

If the curve coefficients are too general to fit any of the above domain categories, only basic operations, such as point addition, will be supported later.

If the curve (seen over the domain D) is singular, fail and return an empty vector [].

? E = ellinit([0,0,0,0,1]); \\ y^2 = x^3 + 1, over Q ? E = ellinit([0,1]); \\ the same curve, short form ? E = ellinit("36a1"); \\ sill the same curve, Cremona's notations ? E = ellinit([0,1], 2) \\ over F2: singular curve %4 = [] ? E = ellinit(['a4,'a6] * Mod(1,5)); \\ over F_5[a4,a6], basic support !

The result of `ellinit`

is an *ell* structure. It contains at least
the following information in its components:

a_1,a_2,a_3,a_4,a_6,b_2,b_4,b_6,b_8,c_4,c_6,Δ,j.

All are accessible via member functions. In particular, the discriminant is
`E.disc`

, and the j-invariant is `E.j`

.

? E = ellinit([a4, a6]); ? E.disc %2 = -64*a4^3 - 432*a6^2 ? E.j %3 = -6912*a4^3/(-4*a4^3 - 27*a6^2)

Further components contain domain-specific data, which are in general dynamic: only computed when needed, and then cached in the structure.

? E = ellinit([2,3], 10^60+7); \\ E over F_p, p large ? ellap(E) time = 4,440 ms. %2 = -1376268269510579884904540406082 ? ellcard(E); \\ now instantaneous ! time = 0 ms. ? ellgenerators(E); time = 5,965 ms. ? ellgenerators(E); \\ second time instantaneous time = 0 ms.

See the description of member functions related to elliptic curves at the beginning of this section.

The library syntax is `GEN `

.**ellinit**(GEN x, GEN D = NULL, long prec)

Let E be an `ell`

structure over a number field K. This function
returns an integral model. If v is present, sets v = [u,0,0,0] to the
corresponding change of variable: the return value is identical to that of
`ellchangecurve(E, v)`

.

The library syntax is `GEN `

.**ellintegralmodel**(GEN E, GEN *v = NULL)

Given E/K a number field and P in E(K)
return 1 if P = [n]R for some R in E(K) and set Q to one such R;
and return 0 otherwise. The integer n ≥ 0 may be given as
`ellxn(E,n)`

, if many points need to be tested.

? K = nfinit(polcyclo(11,t)); ? E = ellinit([0,-1,1,0,0], K); ? P = [0,0]; ? ellorder(E,P) %4 = 5 ? ellisdivisible(E,P,5, &Q) %5 = 1 ? lift(Q) %6 = [-t^7-t^6-t^5-t^4+1, -t^9-2*t^8-2*t^7-3*t^6-3*t^5-2*t^4-2*t^3-t^2-1] ? ellorder(E, Q) %7 = 25

The algebraic complexity of the underlying algorithm is in
O(n^4), so it is advisable to first factor n, then use a chain of checks
attached to the prime divisors of n: the function will do it itself unless
n is given in `ellxn`

form.

The library syntax is `long `

.**ellisdivisible**(GEN E, GEN P, GEN n, GEN *Q) = NULL)

Given an elliptic curve E, a finite subgroup G of E is given either
as a generating point P (for a cyclic G) or as a polynomial whose roots
vanish on the x-coordinates of the non-zero elements of G (general case
and more efficient if available). This function returns the
[a_1,a_2,a_3,a_4,a_6] invariants of the quotient elliptic curve E/G and
(if *only_image* is zero (the default)) a vector of rational
functions [f, g, h] such that the isogeny E → E/G is given by (x,y)
` ⟼ `

(f(x)/h(x)^2, g(x,y)/h(x)^3).

? E = ellinit([0,1]); ? elltors(E) %2 = [6, [6], [[2, 3]]] ? ellisogeny(E, [2,3], 1) \\ Weierstrass model for E/<P> %3 = [0, 0, 0, -135, -594] ? ellisogeny(E,[-1,0]) %4 = [[0,0,0,-15,22], [x^3+2*x^2+4*x+3, y*x^3+3*y*x^2-2*y, x+1]]

The library syntax is `GEN `

where **ellisogeny**(GEN E, GEN G, long only_image, long x = -1, long y = -1)`x`

, `y`

are variable numbers.

Given an isogeny of elliptic curves f:E' → E (being the result of a call
to `ellisogeny`

), apply f to g:

***** if g is a point P in the domain of f, return the image f(P);

***** if g:E" → E' is a compatible isogeny, return the composite
isogeny f o g: E" → E.

? one = ffgen(101, 't)^0; ? E = ellinit([6, 53, 85, 32, 34] * one); ? P = [84, 71] * one; ? ellorder(E, P) %4 = 5 ? [F, f] = ellisogeny(E, P); \\ f: E->F = E/<P> ? ellisogenyapply(f, P) %6 = [0] ? F = ellinit(F); ? Q = [89, 44] * one; ? ellorder(F, Q) %9 = 2 ? [G, g] = ellisogeny(F, Q); \\ g: F->G = F/<Q> ? gof = ellisogenyapply(g, f); \\ gof: E -> G

The library syntax is `GEN `

.**ellisogenyapply**(GEN f, GEN g)

Given an elliptic curve E defined over ℚ, compute representatives of the
isomorphism classes of elliptic curves ℚ-isogenous to E. The function
returns a vector [L,M] where L is a list of triples [E_i, f_i, g_i],
where E_i is an elliptic curve in [a_4,a_6] form, f_i: E → E_i
is a rational isogeny, g_i: E_i → E is the dual isogeny of f_i,
and M is the matrix such that M_{i,j} is the degree of the isogeny between
E_i and E_j. Furthermore the first curve E_1 is isomorphic to E
by f_1. If the flag *fl* = 1, the f_i and g_i are not computed,
which saves time, and L is the list of the curves E_i.

? E = ellinit("14a1"); ? [L,M] = ellisomat(E); ? LE = apply(x->x[1], L) \\ list of curves %3 = [[215/48,-5291/864],[-675/16,6831/32],[-8185/48,-742643/864], [-1705/48,-57707/864],[-13635/16,306207/32],[-131065/48,-47449331/864]] ? L[2][2] \\ isogeny f_2 %4 = [x^3+3/4*x^2+19/2*x-311/12, 1/2*x^4+(y+1)*x^3+(y-4)*x^2+(-9*y+23)*x+(55*y+55/2),x+1/3] ? L[2][3] \\ dual isogeny g_2 %5 = [1/9*x^3-1/4*x^2-141/16*x+5613/64, -1/18*x^4+(1/27*y-1/3)*x^3+(-1/12*y+87/16)*x^2+(49/16*y-48)*x +(-3601/64*y+16947/512),x-3/4] ? apply(E->ellidentify(ellinit(E))[1][1], LE) %6 = ["14a1","14a4","14a3","14a2","14a6","14a5"] ? M %7 = [1 3 3 2 6 6] [3 1 9 6 2 18] [3 9 1 6 18 2] [2 6 6 1 3 3] [6 2 18 3 1 9] [6 18 2 3 9 1]

The library syntax is `GEN `

.**ellisomat**(GEN E, long fl)

Gives 1 (i.e. true) if the point z is on the elliptic curve E, 0 otherwise. If E or z have imprecise coefficients, an attempt is made to take this into account, i.e. an imprecise equality is checked, not a precise one. It is allowed for z to be a vector of points in which case a vector (of the same type) is returned.

The library syntax is `GEN `

.
Also available is **ellisoncurve**(GEN E, GEN z)`int `

which does not
accept vectors of points.**oncurve**(GEN E, GEN z)

Return 1 if the elliptic curve E defined over a number field or a finite field is supersingular at p, and 0 otherwise. If the curve is defined over a number field, p must be explicitly given, and must be a prime number, resp. a maximal ideal, if the curve is defined over ℚ, resp. a general number field: we return 1 if and only if E has supersingular good reduction at p.

Alternatively, E can be given by its j-invariant in a finite field. In this case p must be omitted.

? g = ffprimroot(ffgen(7^5)) %1 = x^3 + 2*x^2 + 3*x + 1 ? [g^n | n <- [1 .. 7^5 - 1], ellissupersingular(g^n)] %2 = [6] ? K = nfinit(y^3-2); P = idealprimedec(K, 2)[1]; ? E = ellinit([y,1], K); ? ellissupersingular(E, P) %5 = 1

The library syntax is `GEN `

.
Also available is
**ellissupersingular**(GEN E, GEN p = NULL)`int `

where j is a j-invariant of a curve
over a finite field.**elljissupersingular**(GEN j)

Elliptic j-invariant. x must be a complex number with positive imaginary part, or convertible into a power series or a p-adic number with positive valuation.

The library syntax is `GEN `

.**jell**(GEN x, long prec)

Calculates the Kodaira type of the local fiber of the elliptic curve
E at p. E must be an `ell`

structure as output by
`ellinit`

, over ℚ (p a rational prime) or a number field K (p
a maximal ideal given by a `prid`

structure), and is assumed to have all
its coefficients a_i integral.
The result is a 4-component vector [f,kod,v,c]. Here f is the exponent of
p in the arithmetic conductor of E, and kod is the Kodaira type which
is coded as follows:

1 means good reduction (type I_0), 2, 3 and 4 mean types II, III and IV respectively, 4+ν with ν > 0 means type I_ν; finally the opposite values -1, -2, etc. refer to the starred types I_0^*, II^*, etc. The third component v is itself a vector [u,r,s,t] giving the coordinate changes done during the local reduction; u = 1 if and only if the given equation was already minimal at p. Finally, the last component c is the local Tamagawa number c_p.

The library syntax is `GEN `

.**elllocalred**(GEN E, GEN p)

Given two points P and G on the elliptic curve E/𝔽_q, returns the
discrete logarithm of P in base G, i.e. the smallest non-negative
integer n such that P = [n]G.
See `znlog`

for the limitations of the underlying discrete log algorithms.
If present, o represents the order of G, see Section se:DLfun;
the preferred format for this parameter is `[N, factor(N)]`

, where N
is the order of G.

If no o is given, assume that G generates the curve. The function also assumes that P is a multiple of G.

? a = ffgen(ffinit(2,8),'a); ? E = ellinit([a,1,0,0,1]); \\ over F_{2^8} ? x = a^3; y = ellordinate(E,x)[1]; ? P = [x,y]; G = ellmul(E, P, 113); ? ord = [242, factor(242)]; \\ P generates a group of order 242. Initialize. ? ellorder(E, G, ord) %4 = 242 ? e = elllog(E, P, G, ord) %5 = 15 ? ellmul(E,G,e) == P %6 = 1

The library syntax is `GEN `

.**elllog**(GEN E, GEN P, GEN G, GEN o = NULL)

This function is deprecated, use `lfun(E,s)`

instead.

E being an elliptic curve, given by an arbitrary model over ℚ as output
by `ellinit`

, this function computes the value of the L-series of E at
the (complex) point s. This function uses an O(N^{1/2}) algorithm, where
N is the conductor.

The optional parameter A fixes a cutoff point for the integral and is best
left omitted; the result must be independent of A, up to
`realprecision`

, so this allows to check the function's accuracy.

The library syntax is `GEN `

.**elllseries**(GEN E, GEN s, GEN A = NULL, long prec)

Let E be an `ell`

structure over a number field K. This function
determines whether E admits a global minimal integral model. If so, it
returns it and sets v = [u,r,s,t] to the corresponding change of variable:
the return value is identical to that of `ellchangecurve(E, v)`

.

Else return the (non-principal) Weierstrass class of E, i.e. the class of
∏ 𝔭^{(v_{𝔭}{Δ} - δ_{𝔭}) / 12} where
Δ = `E.disc`

is the model's discriminant and
𝔭 ^ δ_{𝔭} is the local minimal discriminant.
This function requires either that E be defined
over the rational field ℚ (with domain D = 1 in `ellinit`

),
in which case a global minimal model always exists, or over a number
field given by a *bnf* structure. The Weierstrass class is given in
`bnfisprincipal`

format, i.e. in terms of the `K.gen`

generators.

The resulting model has integral coefficients and is everywhere minimal, the
coefficients a_1 and a_3 are reduced modulo 2 (in terms of the fixed
integral basis `K.zk`

) and a_2 is reduced modulo 3. Over ℚ, we
further require that a_1 and a_3 be 0 or 1, that a_2 be 0 or ±
1 and that u > 0 in the change of variable: both the model and the change
of variable v are then unique.

? e = ellinit([6,6,12,55,233]); \\ over Q ? E = ellminimalmodel(e, &v); ? E[1..5] %3 = [0, 0, 0, 1, 1] ? v %4 = [2, -5, -3, 9]

? K = bnfinit(a^2-65); \\ over a non-principal number field ? K.cyc %2 = [2] ? u = Mod(8+a, K.pol); ? E = ellinit([1,40*u+1,0,25*u^2,0], K); ? ellminimalmodel(E) \\ no global minimal model exists over Z_K %6 = [1]~

The library syntax is `GEN `

.**ellminimalmodel**(GEN E, GEN *v = NULL)

Let E be an elliptic curve defined over ℚ, return
a discriminant D such that the twist of E by D is minimal among all
possible quadratic twists, i.e. if *flag* = 0, its minimal model has minimal
discriminant, or if *flag* = 1, it has minimal conductor.

In the example below, we find a curve with j-invariant 3 and minimal conductor.

? E=ellminimalmodel(ellinit(ellfromj(3))); ? ellglobalred(E)[1] %2 = 357075 ? D = ellminimaltwist(E,1) %3 = -15 ? E2=ellminimalmodel(ellinit(elltwist(E,D))); ? ellglobalred(E2)[1] %5 = 14283

The library syntax is `GEN `

.
Also available are
**ellminimaltwist0**(GEN E, long flag)`GEN `

for **ellminimaltwist**(E)*flag* = 0, and
`GEN `

for **ellminimaltwistcond**(E)*flag* = 1.

e being an elliptic curve defined over ℚ output by `ellinit`

,
compute the modular degree of e divided by the square of
the Manin constant. Return [D, err], where D is a rational number and
err is exponent of the truncation error.

The library syntax is `GEN `

.**ellmoddegree**(GEN e, long bitprec)

Given a prime N < 500, return a vector [P,t] where P(x,y)
is a modular equation of level N, i.e. a bivariate polynomial with integer
coefficients; t indicates the type of this equation: either
*canonical* (t = 0) or *Atkin* (t = 1). This function requires
the `seadata`

package and its only use is to give access to the package
contents. See `polmodular`

for a more general and more flexible function.

Let j be the j-invariant function. The polynomial P satisfies the functional equation, P(f,j) = P(f | W_N, j | W_N) = 0 for some modular function f = f_N (hand-picked for each fixed N to minimize its size, see below), where W_N(τ) = -1 / (N τ) is the Atkin-Lehner involution. These two equations allow to compute the values of the classical modular polynomial Φ_N, such that Φ_N(j(τ), j(Nτ)) = 0, while being much smaller than the latter. More precisely, we have j(W_N(τ)) = j(N τ); the function f is invariant under Γ_0(N) and also satisfies

***** for Atkin type: f | W_N = f;

***** for canonical type: let s = 12/gcd(12,N-1), then
f | W_N = N^s / f. In this case, f has a simple definition:
f(τ) = N^s (η(N τ) / η(τ) )^{2 s},
where η is Dedekind's eta function.

The following GP function returns values of the classical modular polynomial by eliminating f_N(τ) in the above functional equation, for N ≤ 31 or N ∈ {41,47,59,71}.

classicaleqn(N, X='X, Y='Y)= { my([P,t] = ellmodulareqn(N), Q, d); if (poldegree(P,'y) > 2, error("level unavailable in classicaleqn")); if (t == 0, \\ Canonical my(s = 12/gcd(12,N-1)); Q = 'x^(N+1) * substvec(P,['x,'y],[N^s/'x,Y]); d = N^(s*(2*N+1)) * (-1)^(N+1); , \\ Atkin Q = subst(P,'y,Y); d = (X-Y)^(N+1)); polresultant(subst(P,'y,X), Q) / d; }

The library syntax is `GEN `

where **ellmodulareqn**(long N, long x = -1, long y = -1)`x`

, `y`

are variable numbers.

Computes [n]z, where z is a point on the elliptic curve E. The exponent n is in ℤ, or may be a complex quadratic integer if the curve E has complex multiplication by n (if not, an error message is issued).

? Ei = ellinit([1,0]); z = [0,0]; ? ellmul(Ei, z, 10) %2 = [0] \\ unsurprising: z has order 2 ? ellmul(Ei, z, I) %3 = [0, 0] \\ Ei has complex multiplication by Z[i] ? ellmul(Ei, z, quadgen(-4)) %4 = [0, 0] \\ an alternative syntax for the same query ? Ej = ellinit([0,1]); z = [-1,0]; ? ellmul(Ej, z, I) *** at top-level: ellmul(Ej,z,I) *** ^-------------- *** ellmul: not a complex multiplication in ellmul. ? ellmul(Ej, z, 1+quadgen(-3)) %6 = [1 - w, 0]

The simple-minded algorithm for the CM case assumes that we are in characteristic 0, and that the quadratic order to which n belongs has small discriminant.

The library syntax is `GEN `

.**ellmul**(GEN E, GEN z, GEN n)

Opposite of the point z on elliptic curve E.

The library syntax is `GEN `

.**ellneg**(GEN E, GEN z)

Given an elliptic curve E/ℚ (more precisely, a model defined over ℚ of a curve) and a rational point P ∈ E(ℚ), returns the pair [R,n], where n is the least positive integer such that R := [n]P has good reduction at every prime. More precisely, its image in a minimal model is everywhere non-singular.

? e = ellinit("57a1"); P = [2,-2]; ? ellnonsingularmultiple(e, P) %2 = [[1, -1], 2] ? e = ellinit("396b2"); P = [35, -198]; ? [R,n] = ellnonsingularmultiple(e, P); ? n %5 = 12

The library syntax is `GEN `

.**ellnonsingularmultiple**(GEN E, GEN P)

Gives the order of the point z on the elliptic curve E, defined over a finite field or a number field. Return (the impossible value) zero if the point has infinite order.

? E = ellinit([-157^2,0]); \\ the "157-is-congruent" curve ? P = [2,2]; ellorder(E, P) %2 = 2 ? P = ellheegner(E); ellorder(E, P) \\ infinite order %3 = 0 ? K = nfinit(polcyclo(11,t)); E=ellinit("11a3", K); T = elltors(E); ? ellorder(E, T.gen[1]) %5 = 25 ? E = ellinit(ellfromj(ffgen(5^10))); ? ellcard(E) %7 = 9762580 ? P = random(E); ellorder(E, P) %8 = 4881290 ? p = 2^160+7; E = ellinit([1,2], p); ? N = ellcard(E) %9 = 1461501637330902918203686560289225285992592471152 ? o = [N, factor(N)]; ? for(i=1,100, ellorder(E,random(E))) time = 260 ms.

The parameter o, is now mostly useless, and kept for backward
compatibility. If present, it represents a non-zero multiple of the order
of z, see Section se:DLfun; the preferred format for this parameter is
`[ord, factor(ord)]`

, where `ord`

is the cardinality of the curve.
It is no longer needed since PARI is now able to compute it over large
finite fields (was restricted to small prime fields at the time this feature
was introduced), *and* caches the result in E so that it is computed
and factored only once. Modifying the last example, we see that including
this extra parameter provides no improvement:

? o = [N, factor(N)]; ? for(i=1,100, ellorder(E,random(E),o)) time = 260 ms.

The library syntax is `GEN `

.
The obsolete form **ellorder**(GEN E, GEN z, GEN o = NULL)`GEN `

should no longer be
used.**orderell**(GEN e, GEN z)

Gives a 0, 1 or 2-component vector containing the y-coordinates of the points of the curve E having x as x-coordinate.

The library syntax is `GEN `

.**ellordinate**(GEN E, GEN x, long prec)

Returns the value (or r-th derivative) on a character χ^s of ℤ_p^* of the p-adic L-function of the elliptic curve E/ℚ, twisted by D, given modulo p^n.

**Characters.** The set of continuous characters of
Gal(ℚ(μ_{p^{ oo }})/ ℚ) is identified to ℤ_p^* via the
cyclotomic character χ with values in ℚ_p^*. Denote by
τ:ℤ_p^* → ℤ_p^* the Teichmüller character, with values
in the (p-1)-th roots of 1 for p != 2, and {-1,1} for p = 2;
finally, let
`<`

χ`>`

= χ τ^{-1}, with values in 1 + 2pℤ_p.
In GP, the continuous character of
Gal(ℚ(μ_{p^{ oo }})/ ℚ) given by `<`

χ`>`

^{s_1}
τ^{s_2} is represented by the pair of integers s = (s_1,s_2), with s_1
∈ ℤ_p and s_2 mod p-1 for p > 2, (resp. mod 2 for p = 2); s
may be also an integer, representing (s,s) or χ^s.

**The p-adic L function.**
The p-adic L function L_p is defined on the set of continuous
characters of Gal(ℚ(μ_{p^{ oo }})/ ℚ), as ∫_{ℤ_p^*}
χ^s d μ for a certain p-adic distribution μ on ℤ_p^*. The
derivative is given by
L_p^{(r)}(E, χ^s) = ∫_{ℤ_p^*} log_p^r(a) χ^s(a) dμ(a).
More precisely:

***** When E has good supersingular reduction, L_p takes its
values in ℚ_p ⨂ H^1_{dR}(E/ℚ) and satisfies
(1-p^{-1} F)^{-2} L_p(E, χ^0) = (L(E,1) / Ω).ω
where F is the Frobenius, L(E,1) is the value of the complex L
function at 1, ω is the Néron differential
and Ω the attached period on E(ℝ). Here, χ^0 represents
the trivial character.

The function returns the components of L_p^{(r)}(E,χ^s) in the basis (ω, F(ω)).

***** When E has ordinary good reduction, this method only defines
the projection of L_p(E,χ^s) on the α-eigenspace,
where α is the unit eigenvalue for F. This is what the function
returns. We have
(1- α^{-1})^{-2} L_{p,α}(E,χ^0) = L(E,1) / Ω.

Two supersingular examples:

? cxL(e) = bestappr( ellL1(e) / e.omega[1] ); ? e = ellinit("17a1"); p=3; \\ supersingular, a3 = 0 ? L = ellpadicL(e,p,4); ? F = [0,-p;1,ellap(e,p)]; \\ Frobenius matrix in the basis (omega,F(omega)) ? (1-p^(-1)*F)^-2 * L / cxL(e) %5 = [1 + O(3^5), O(3^5)]~ \\ [1,0]~ ? e = ellinit("116a1"); p=3; \\ supersingular, a3 != 0~ ? L = ellpadicL(e,p,4); ? F = [0,-p; 1,ellap(e,p)]; ? (1-p^(-1)*F)^-2*L~ / cxL(e) %9 = [1 + O(3^4), O(3^5)]~

Good ordinary reduction:

? e = ellinit("17a1"); p=5; ap = ellap(e,p) %1 = -2 \\ ordinary ? L = ellpadicL(e,p,4) %2 = 4 + 3*5 + 4*5^2 + 2*5^3 + O(5^4) ? al = padicappr(x^2 - ap*x + p, ap + O(p^7))[1]; ? (1-al^(-1))^(-2) * L / cxL(e) %4 = 1 + O(5^4)

Twist and Teichmüller:

? e = ellinit("17a1"); p=5; \\ ordinary \\ 2nd derivative at tau^1, twist by -7 ? ellpadicL(e, p, 4, [0,1], 2, -7) %2 = 2*5^2 + 5^3 + O(5^4)

This function is a special case of `mspadicL`

, and it also appears
as the first term of `mspadicseries`

:

? e = ellinit("17a1"); p=5; ? L = ellpadicL(e,p,4) %2 = 4 + 3*5 + 4*5^2 + 2*5^3 + O(5^4) ? [M,phi] = msfromell(e, 1); ? Mp = mspadicinit(M, p, 4); ? mu = mspadicmoments(Mp, phi); ? mspadicL(mu) %6 = 4 + 3*5 + 4*5^2 + 2*5^3 + 2*5^4 + 5^5 + O(5^6) ? mspadicseries(mu) %7 = (4 + 3*5 + 4*5^2 + 2*5^3 + 2*5^4 + 5^5 + O(5^6)) + (3 + 3*5 + 5^2 + 5^3 + O(5^4))*x + (2 + 3*5 + 5^2 + O(5^3))*x^2 + (3 + 4*5 + 4*5^2 + O(5^3))*x^3 + (3 + 2*5 + O(5^2))*x^4 + O(x^5)

These are more cumbersome than `ellpadicL`

but allow to
compute at different characters, or successive derivatives, or to
twist by a quadratic character essentially for the cost of a single call to
`ellpadicL`

due to precomputations.

The library syntax is `GEN `

.**ellpadicL**(GEN E, GEN p, long n, GEN s = NULL, long r, GEN D = NULL)

If p > 2 is a prime and E is a elliptic curve on ℚ with good reduction at p, return the matrix of the Frobenius endomorphism ϕ on the crystalline module D_p(E) = ℚ_p ⨂ H^1_{dR}(E/ℚ) with respect to the basis of the given model (ω, η = x ω), where ω = dx/(2 y+a_1 x+a_3) is the invariant differential. The characteristic polynomial of ϕ is x^2 - a_p x + p. The matrix is computed to absolute p-adic precision p^n.

? E = ellinit([1,-1,1,0,0]); ? F = ellpadicfrobenius(E,5,3); ? lift(F) %3 = [120 29] [ 55 5] ? charpoly(F) %4 = x^2 + O(5^3)*x + (5 + O(5^3)) ? ellap(E, 5) %5 = 0

The library syntax is `GEN `

.**ellpadicfrobenius**(GEN E, long p, long n)

Cyclotomic p-adic height of the rational point P on the elliptic curve E (defined over ℚ), given to n p-adic digits. If the argument Q is present, computes the value of the bilinear form (h(P+Q)-h(P-Q)) / 4.

Let D_{dR}(E) := H^1_{dR}(E) ⨂ _ℚ ℚ_p be the ℚ_p vector space spanned by ω (invariant differential dx/(2y+a_1x+a3) related to the given model) and η = x ω. Then the cyclotomic p-adic height associates to P ∈ E(ℚ) an element f ω + gη in D_{dR}. This routine returns the vector [f, g] to n p-adic digits.

If P ∈ E(ℚ) is in the kernel of reduction mod p and if its reduction at all finite places is non singular, then g = -(log_E P)^2, where log_E is the logarithm for the formal group of E at p.

If furthermore the model is of the form Y^2 = X^3 + a X + b and P = (x,y),
then
f = log_p(`denominator`

(x)) - 2 log_p(σ(P))
where σ(P) is given by `ellsigma`

(E,P).

Recall (*Advanced topics in the arithmetic of elliptic
curves*, Theorem 3.2) that the local height function over the complex numbers
is of the form
λ(z) = -log (|`E.disc`

|) / 6 + Re(z η(z)) - 2 log(
σ(z).
(N.B. our normalization for local and global heights is twice that of
Silverman's).

? E = ellinit([1,-1,1,0,0]); P = [0,0]; ? ellpadicheight(E,5,4, P) %2 = [3*5 + 5^2 + 2*5^3 + O(5^4), 5^2 + 4*5^4 + O(5^6)] ? E = ellinit("11a1"); P = [5,5]; \\ torsion point ? ellpadicheight(E,19,6, P) %4 = O(19^6) ? E = ellinit([0,0,1,-4,2]); P = [-2,1]; ? ellpadicheight(E,3,5, P) %6 = [2*3^2 + 2*3^3 + 3^4 + O(3^5), 2*3^2 + 3^4 + 2*3^5 + 3^6 + O(3^7)] ? ellpadicheight(E,3,5, P, elladd(E,P,P))

One can replace the parameter p prime by a vector [p,[a,b]], in which case the routine returns the p-adic number af + bg.

When E has good ordinary reduction at p, the "canonical" p-adic height is given by

s2 = ellpadics2(E,p,n); ellpadicheight(E, [p,[1,-s2]], n, P)

Since s_2 does not depend on P, it is preferable to compute it only once:

? E = ellinit("5077a1"); p = 5; n = 7; ? s2 = ellpadics2(E,p,n); ? M = ellpadicheightmatrix(E,[p,[1,-s2]], n, E.gen); ? matdet(M) \\ p-adic regulator %4 = 5 + 5^2 + 4*5^3 + 2*5^4 + 2*5^5 + 5^6 + O(5^7)

The library syntax is `GEN `

.**ellpadicheight0**(GEN E, GEN p, long n, GEN P, GEN Q = NULL)

v being a vector of points, this function outputs the Gram matrix of
v with respect to the cyclotomic p-adic height, given to n p-adic
digits; in other words, the (i,j) component of the matrix is equal to
`ellpadicheight`

(E,p,n, v[i],v[j]) = [f,g].

See `ellpadicheight`

; in particular one can replace the parameter p
prime by a vector [p,[a,b]], in which case the routine returns the matrix
containing the p-adic numbers af + bg.

The library syntax is `GEN `

.**ellpadicheightmatrix**(GEN E, GEN p, long n, GEN v)

Given E defined over K = ℚ or ℚ_p and P = [x,y] on E(K) in the kernel of reduction mod p, let t(P) = -x/y be the formal group parameter; this function returns L(t), where L denotes the formal logarithm (mapping the formal group of E to the additive formal group) attached to the canonical invariant differential: dL = dx/(2y + a_1x + a_3).

The library syntax is `GEN `

.**ellpadiclog**(GEN E, GEN p, long n, GEN P)

If p > 2 is a prime and E/ℚ is a elliptic curve with ordinary good
reduction at p, returns the slope of the unit eigenvector
of `ellpadicfrobenius(E,p,n)`

, i.e. the action of Frobenius ϕ on
the crystalline module D_p(E) = ℚ_p ⨂ H^1_{dR}(E/ℚ) in the basis of
the given model (ω, η = x ω), where ω is the invariant
differential dx/(2 y+a_1 x+a_3). In other words, η + s_2ω
is an eigenvector for the unit eigenvalue of ϕ.

This slope is the unique c ∈ 3^{-1}ℤ_p such that the odd solution σ(t) = t + O(t^2) of - d((1)/(σ) (d σ)/(ω)) = (x(t) + c) ω is in tℤ_p[[t]].

It is equal to b_2/12 - E_2/12 where E_2 is the value of the Katz p-adic Eisenstein series of weight 2 on (E,ω). This is used to construct a canonical p-adic height when E has good ordinary reduction at p as follows

s2 = ellpadics2(E,p,n); h(E,p,n, P, s2) = ellpadicheight(E, [p,[1,-s2]],n, P);

Since s_2 does not depend on the point P, we compute it only once.

The library syntax is `GEN `

.**ellpadics2**(GEN E, GEN p, long n)

Let w describe a complex period lattice (w = [w_1,w_2]
or an `ellinit`

structure). Returns normalized periods [W_1,W_2] generating
the same lattice such that τ := W_1/W_2 has positive imaginary part
and lies in the standard fundamental domain for SL_2(ℤ).

If *flag* = 1, the function returns [[W_1,W_2], [η_1,η_2]], where
η_1 and η_2 are the quasi-periods attached to
[W_1,W_2], satisfying η_1 W_2 - η_2 W_1 = 2 i π.

The output of this function is meant to be used as the first argument given to ellwp, ellzeta, ellsigma or elleisnum. Quasi-periods are needed by ellzeta and ellsigma only.

The library syntax is `GEN `

.**ellperiods**(GEN w, long flag, long prec)

If E/ℂ ~ ℂ/Λ is a complex elliptic curve (Λ =
`E.omega`

),
computes a complex number z, well-defined modulo the lattice Λ,
corresponding to the point P; i.e. such that
P = [℘_Λ(z),℘'_Λ(z)]
satisfies the equation
y^2 = 4x^3 - g_2 x - g_3,
where g_2, g_3 are the elliptic invariants.

If E is defined over ℝ and P ∈ E(ℝ), we have more precisely, 0 ≤ Re(t) < w1 and 0 ≤ Im(t) < Im(w2), where (w1,w2) are the real and complex periods of E.

? E = ellinit([0,1]); P = [2,3]; ? z = ellpointtoz(E, P) %2 = 3.5054552633136356529375476976257353387 ? ellwp(E, z) %3 = 2.0000000000000000000000000000000000000 ? ellztopoint(E, z) - P %4 = [6.372367644529809109 E-58, 7.646841173435770930 E-57] ? ellpointtoz(E, [0]) \\ the point at infinity %5 = 0

If E/ℚ_p has multiplicative reduction, then E/ℚ_p is analytically isomorphic to ℚ_p^*/q^ℤ (Tate curve) for some p-adic integer q. The behaviour is then as follows:

***** If the reduction is split (E.`tate[2]`

is a `t_PADIC`

), we have
an isomorphism φ: E(ℚ_p) ~ ℚ_p^*/q^ℤ and the function returns
φ(P) ∈ ℚ_p.

***** If the reduction is *not* split (E.`tate[2]`

is a
`t_POLMOD`

), we only have an isomorphism φ: E(K) ~ K^*/q^ℤ over
the unramified quadratic extension K/ℚ_p. In this case, the output
φ(P) ∈ K is a `t_POLMOD`

.

? E = ellinit([0,-1,1,0,0], O(11^5)); P = [0,0]; ? [u2,u,q] = E.tate; type(u) \\ split multiplicative reduction %2 = "t_PADIC" ? ellmul(E, P, 5) \\ P has order 5 %3 = [0] ? z = ellpointtoz(E, [0,0]) %4 = 3 + 11^2 + 2*11^3 + 3*11^4 + O(11^5) ? z^5 %5 = 1 + O(11^5) ? E = ellinit(ellfromj(1/4), O(2^6)); x=1/2; y=ellordinate(E,x)[1]; ? z = ellpointtoz(E,[x,y]); \\ t_POLMOD of t_POL with t_PADIC coeffs ? liftint(z) \\ lift all p-adics %8 = Mod(8*u + 7, u^2 + 437)

The library syntax is `GEN `

.**zell**(GEN E, GEN P, long prec)

Deprecated alias for `ellmul`

.

The library syntax is `GEN `

.**ellmul**(GEN E, GEN z, GEN n)

E being an `ell`

structure over ℚ as output by `ellinit`

,
this function computes the local root number of its L-series at the place
p (at the infinite place if p = 0). If p is omitted, return the global
root number. Note that the global root number is the sign of the functional
equation and conjecturally is the parity of the rank of the
Mordell-Weil group. The equation for E needs not be minimal at p,
but if the model is already minimal the function will run faster.

The library syntax is `long `

.**ellrootno**(GEN E, GEN p = NULL)

Let E be an *ell* structure as output by `ellinit`

, defined over
a finite field 𝔽_q. This function computes the order of the group
E(𝔽_q) using the SEA algorithm and the `tors`

argument allows to
speed up a search for curves having almost prime order.

***** If the characteristic is too small (p ≤ 7) the generic algorithm
`ellcard`

is used instead and the `tors`

argument is ignored.

***** When `tors`

is set to a non-zero value, the function returns 0
as soon as it detects that the order has a small prime factor not dividing
`tors`

; SEA considers modular polynomials of increasing prime degree
ℓ and we return 0 as soon as we hit an ℓ (coprime to `tors`

)
dividing #E(𝔽_q).

In particular, you should set `tors`

to 1 if you want a curve with
prime order, to 2 if you want to allow a cofacteur which is a power of two
(e.g. for Edwards's curves), etc.

The availability of the `seadata`

package will speed up the computation,
and is strongly recommended.

The following function returns a curve of prime order over 𝔽_p.

cryptocurve(p) = { while(1, my(E, N, j = Mod(random(p), p)); E = ellinit(ellfromj(j)); N = ellsea(E, 1); if(!N, continue); if (isprime(N), return(E)); \\ try the quadratic twist for free if (isprime(2*p+2 - N), return(ellinit(elltwist(E)))); ); } ? p = randomprime([2^255, 2^256]); ? E = cryptocurve(p); \\ insist on prime order %2 = 47,447ms

The same example without early abort (using `ellsea(E,1)`

instead of `ellsea(E)`

) runs for about 5 minutes before finding a
suitable curve.

The library syntax is `GEN `

.**ellsea**(GEN E, ulong tors)

This function finds all curves in the `elldata`

database satisfying
the constraint defined by the argument N:

***** if N is a character string, it selects a given curve, e.g.
`"11a1"`

, or curves in the given isogeny class, e.g. `"11a"`

, or
curves with given conductor, e.g. `"11"`

;

***** if N is a vector of integers, it encodes the same constraints
as the character string above, according to the `ellconvertname`

correspondance, e.g. `[11,0,1]`

for `"11a1"`

, `[11,0]`

for
`"11a"`

and `[11]`

for `"11"`

;

***** if N is an integer, curves with conductor N are selected.

If N codes a full curve name, for instance `"11a1"`

or `[11,0,1]`

,
the output format is [N, [a_1,a_2,a_3,a_4,a_6], G] where
[a_1,a_2,a_3,a_4,a_6] are the coefficients of the Weierstrass equation of
the curve and G is a ℤ-basis of the free part of the
Mordell-Weil group attached to the curve.

? ellsearch("11a3") %1 = ["11a3", [0, -1, 1, 0, 0], []] ? ellsearch([11,0,3]) %2 = ["11a3", [0, -1, 1, 0, 0], []]

If N is not a full curve name, then the output is a vector of all matching curves in the above format:

? ellsearch("11a") %1 = [["11a1", [0, -1, 1, -10, -20], []], ["11a2", [0, -1, 1, -7820, -263580], []], ["11a3", [0, -1, 1, 0, 0], []]] ? ellsearch("11b") %2 = []

The library syntax is `GEN `

.
Also available is **ellsearch**(GEN N)`GEN `

that only
accepts complete curve names (as **ellsearchcurve**(GEN N)`t_STR`

).

Computes the value at z of the Weierstrass σ function attached to
the lattice L as given by `ellperiods`

(,1): including quasi-periods
is useful, otherwise there are recomputed from scratch for each new z.
σ(z, L) = z ∏_{ω ∈ L^*} (1 -
(z)/(ω))e^{(z)/(ω) + (z^2)/(2ω^2)}.
It is also possible to directly input L = [ω_1,ω_2],
or an elliptic curve E as given by `ellinit`

(L = `E.omega`

).

? w = ellperiods([1,I], 1); ? ellsigma(w, 1/2) %2 = 0.47494937998792065033250463632798296855 ? E = ellinit([1,0]); ? ellsigma(E) \\ at 'x, implicitly at default seriesprecision %4 = x + 1/60*x^5 - 1/10080*x^9 - 23/259459200*x^13 + O(x^17)

If *flag* = 1, computes an arbitrary determination of log(σ(z)).

The library syntax is `GEN `

.**ellsigma**(GEN L, GEN z = NULL, long flag, long prec)

Difference of the points z1 and z2 on the elliptic curve corresponding to E.

The library syntax is `GEN `

.**ellsub**(GEN E, GEN z1, GEN z2)

Computes the modular parametrization of the elliptic curve E/ℚ,
where E is an `ell`

structure as output by `ellinit`

. This returns
a two-component vector [u,v] of power series, given to d significant
terms (`seriesprecision`

by default), characterized by the following two
properties. First the point (u,v) satisfies the equation of the elliptic
curve. Second, let N be the conductor of E and Φ: X_0(N) → E
be a modular parametrization; the pullback by Φ of the
Néron differential du/(2v+a_1u+a_3) is equal to 2iπ
f(z)dz, a holomorphic differential form. The variable used in the power
series for u and v is x, which is implicitly understood to be equal to
exp(2iπ z).

The algorithm assumes that E is a *strong* Weil curve
and that the Manin constant is equal to 1: in fact, f(x) = ∑_{n > 0}
`ellan`

(E, n) x^n.

The library syntax is `GEN `

.**elltaniyama**(GEN E, long precdl)

Computes the Tate pairing of the two points P and Q on the elliptic curve E. The point P must be of m-torsion.

The library syntax is `GEN `

.**elltatepairing**(GEN E, GEN P, GEN Q, GEN m)

If E is an elliptic curve defined over a number field or a finite field,
outputs the torsion subgroup of E as a 3-component vector `[t,v1,v2]`

,
where `t`

is the order of the torsion group, `v1`

gives the structure
of the torsion group as a product of cyclic groups (sorted by decreasing
order), and `v2`

gives generators for these cyclic groups. E must be an
`ell`

structure as output by `ellinit`

.

? E = ellinit([-1,0]); ? elltors(E) %1 = [4, [2, 2], [[0, 0], [1, 0]]]

Here, the torsion subgroup is isomorphic to ℤ/2ℤ x ℤ/2ℤ, with generators [0,0] and [1,0].

The library syntax is `GEN `

.**elltors**(GEN E)

Returns the coefficients [a_1,a_2,a_3,a_4,a_6] of the twist of the
elliptic curve E by the quadratic extension of the coefficient ring
defined by P (when P is a polynomial) or `quadpoly(P)`

when P is an
integer. If E is defined over a finite field, then P can be omitted,
in which case a random model of the unique non-trivial twist is returned.
If E is defined over a number field, the model should be replaced by a
minimal model (if one exists).

Example: Twist by discriminant -3:

? elltwist(ellinit([0,a2,0,a4,a6]),-3) %1 = [0,-3*a2,0,9*a4,-27*a6]

Twist by the Artin-Shreier extension given by x^2+x+T in characteristic 2:

? lift(elltwist(ellinit([a1,a2,a3,a4,a6]*Mod(1,2)),x^2+x+T)) %1 = [a1,a2+a1^2*T,a3,a4,a6+a3^2*T]

Twist of an elliptic curve defined over a finite field:

? E=ellinit([1,7]*Mod(1,19));lift(elltwist(E)) %1 = [0,0,0,11,12]

The library syntax is `GEN `

.**elltwist**(GEN E, GEN P = NULL)

Computes the Weil pairing of the two points of m-torsion P and Q on the elliptic curve E.

The library syntax is `GEN `

.**ellweilpairing**(GEN E, GEN P, GEN Q, GEN m)

Computes the value at z of the Weierstrass ℘ function attached to
the lattice w as given by `ellperiods`

. It is also possible to
directly input w = [ω_1,ω_2], or an elliptic curve E as given
by `ellinit`

(w = `E.omega`

).

? w = ellperiods([1,I]); ? ellwp(w, 1/2) %2 = 6.8751858180203728274900957798105571978 ? E = ellinit([1,1]); ? ellwp(E, 1/2) %4 = 3.9413112427016474646048282462709151389

One can also compute the series expansion around z = 0:

? E = ellinit([1,0]); ? ellwp(E) \\ 'x implicitly at default seriesprecision %5 = x^-2 - 1/5*x^2 + 1/75*x^6 - 2/4875*x^10 + O(x^14) ? ellwp(E, x + O(x^12)) \\ explicit precision %6 = x^-2 - 1/5*x^2 + 1/75*x^6 + O(x^9)

Optional *flag* means 0 (default): compute only ℘(z), 1: compute
[℘(z),℘'(z)].

The library syntax is `GEN `

.
For **ellwp0**(GEN w, GEN z = NULL, long flag, long prec)*flag* = 0, we also have
`GEN `

, and
**ellwp**(GEN w, GEN z, long prec)`GEN `

for the power series in
variable v.**ellwpseries**(GEN E, long v, long precdl)

In standard notation, for any affine point P = (v,w) on the curve E, we have [n]P = (φ_n(P)ψ_n(P) : ω_n(P) : ψ_n(P)^3) for some polynomials φ_n,ω_n,ψ_n in ℤ[a_1,a_2,a_3,a_4,a_6][v,w]. This function returns [φ_n(P),ψ_n(P)^2], which give the numerator and denominator of the abcissa of [n]P and depend only on v.

The library syntax is `GEN `

where **ellxn**(GEN E, long n, long v = -1)`v`

is a variable number.

Computes the value at z of the Weierstrass ζ function attached to
the lattice w as given by `ellperiods`

(,1): including quasi-periods
is useful, otherwise there are recomputed from scratch for each new z.
ζ(z, L) = (1)/(z) + z^2∑_{ω ∈ L^*}
(1)/(ω^2(z-ω)).
It is also possible to directly input w = [ω_1,ω_2],
or an elliptic curve E as given by `ellinit`

(w = `E.omega`

).
The quasi-periods of ζ, such that
ζ(z + aω_1 + bω_2) = ζ(z) + aη_1 + bη_2
for integers a and b are obtained as η_i = 2ζ(ω_i/2).
Or using directly `elleta`

.

? w = ellperiods([1,I],1); ? ellzeta(w, 1/2) %2 = 1.5707963267948966192313216916397514421 ? E = ellinit([1,0]); ? ellzeta(E, E.omega[1]/2) %4 = 0.84721308479397908660649912348219163647

One can also compute the series expansion around z = 0 (the quasi-periods are useless in this case):

? E = ellinit([0,1]); ? ellzeta(E) \\ at 'x, implicitly at default seriesprecision %4 = x^-1 + 1/35*x^5 - 1/7007*x^11 + O(x^15) ? ellzeta(E, x + O(x^20)) \\ explicit precision %5 = x^-1 + 1/35*x^5 - 1/7007*x^11 + 1/1440257*x^17 + O(x^18)

The library syntax is `GEN `

.**ellzeta**(GEN w, GEN z = NULL, long prec)

E being an *ell* as output by
`ellinit`

, computes the coordinates [x,y] on the curve E
corresponding to the complex number z. Hence this is the inverse function
of `ellpointtoz`

. In other words, if the curve is put in Weierstrass
form y^2 = 4x^3 - g_2x - g_3, [x,y] represents the Weierstrass
℘-function and its derivative. More
precisely, we have
x = ℘(z) - b_2/12, y = ℘'(z) - (a_1 x + a_3)/2.
If z is in the lattice defining E over ℂ, the result is the point at
infinity [0].

The library syntax is `GEN `

.**pointell**(GEN E, GEN z, long prec)

Let PQ be a polynomial P, resp. a vector [P,Q] of polynomials, with rational coefficients. Determines the reduction at p > 2 of the (proper, smooth) genus 2 curve C/ℚ, defined by the hyperelliptic equation y^2 = P(x), resp. y^2 + Q(x)*y = P(x). (The special fiber X_p of the minimal regular model X of C over ℤ.)

If p is omitted, determines the reduction type for all (odd) prime divisors of the discriminant.

This function was rewritten from an implementation of Liu's
algorithm by Cohen and Liu (1994), `genus2reduction-0.3`

, see
`http://www.math.u-bordeaux.fr/~liu/G2R/`

.

**CAVEAT.** The function interface may change: for the
time being, it returns [N,*FaN*, T, V]
where N is either the local conductor at p or the
global conductor, *FaN* is its factorization, y^2 = T defines a
minimal model over ℤ[1/2] and V describes the reduction type at the
various considered p. Unfortunately, the program is not complete for
p = 2, and we may return the odd part of the conductor only: this is the
case if the factorization includes the (impossible) term 2^{-1}; if the
factorization contains another power of 2, then this is the exact local
conductor at 2 and N is the global conductor.

? default(debuglevel, 1); ? genus2red(x^6 + 3*x^3 + 63, 3) (potential) stable reduction: [1, []] reduction at p: [III{9}] page 184, [3, 3], f = 10 %1 = [59049, Mat([3, 10]), x^6 + 3*x^3 + 63, [3, [1, []], ["[III{9}] page 184", [3, 3]]]] ? [N, FaN, T, V] = genus2red(x^3-x^2-1, x^2-x); \\ X_1(13), global reduction p = 13 (potential) stable reduction: [5, [Mod(0, 13), Mod(0, 13)]] reduction at p: [I{0}-II-0] page 159, [], f = 2 ? N %3 = 169 ? FaN %4 = Mat([13, 2]) \\ in particular, good reduction at 2 ! ? T %5 = x^6 + 58*x^5 + 1401*x^4 + 18038*x^3 + 130546*x^2 + 503516*x + 808561 ? V %6 = [[13, [5, [Mod(0, 13), Mod(0, 13)]], ["[I{0}-II-0] page 159", []]]]

We now first describe the format of the vector V = V_p in the case where
p was specified (local reduction at p): it is a triple [p, *stable*,
*red*]. The component *stable* = [*type*, *vecj*] contains
information about the stable reduction after a field extension;
depending on *type*s, the stable reduction is

***** 1: smooth (i.e. the curve has potentially good reduction). The
Jacobian J(C) has potentially good reduction.

***** 2: an elliptic curve E with an ordinary double point; *vecj*
contains j mod p, the modular invariant of E. The (potential)
semi-abelian reduction of J(C) is the extension of an elliptic curve (with
modular invariant j mod p) by a torus.

***** 3: a projective line with two ordinary double points. The Jacobian
J(C) has potentially multiplicative reduction.

***** 4: the union of two projective lines crossing transversally at three
points. The Jacobian J(C) has potentially multiplicative reduction.

***** 5: the union of two elliptic curves E_1 and E_2 intersecting
transversally at one point; *vecj* contains their modular invariants
j_1 and j_2, which may live in a quadratic extension of 𝔽_p and need
not be distinct. The Jacobian J(C) has potentially good reduction,
isomorphic to the product of the reductions of E_1 and E_2.

***** 6: the union of an elliptic curve E and a projective line which has
an ordinary double point, and these two components intersect transversally
at one point; *vecj* contains j mod p, the modular invariant of E.
The (potential) semi-abelian reduction of J(C) is the extension of an
elliptic curve (with modular invariant j mod p) by a torus.

***** 7: as in type 6, but the two components are both singular. The
Jacobian J(C) has potentially multiplicative reduction.

The component *red* = [*NUtype*, *neron*] contains two data
concerning the reduction at p without any ramified field extension.

The *NUtype* is a `t_STR`

describing the reduction at p of C,
following Namikawa-Ueno, *The complete classification of fibers in
pencils of curves of genus two*, Manuscripta Math., vol. 9, (1973), pages
143-186. The reduction symbol is followed by the corresponding page number
or page range in this article.

The second datum *neron* is the group of connected components (over an
algebraic closure of 𝔽_p) of the Néron model of J(C), given as a
finite abelian group (vector of elementary divisors).

If p = 2, the *red* component may be omitted altogether (and
replaced by `[]`

, in the case where the program could not compute it.
When p was not specified, V is the vector of all V_p, for all
considered p.

**Notes about Namikawa-Ueno types.**

***** A lower index is denoted between braces: for instance,
`[I{2}-II-5]`

means `[I_2-II-5]`

.

***** If K and K' are Kodaira symbols for singular fibers of elliptic
curves, then `[K-K'-m]`

and `[K'-K-m]`

are the same.

We define a total ordering on Kodaira symbol by fixing `I`

< `I*`

<
`II`

< `II*`

,.... If the reduction type is the same, we order by
the number of components, e.g. `I`

_2 < `I`

_4, etc.
Then we normalize our output so that K ≤ K'.

***** `[K-K'--1]`

is `[K-K'-α]`

in the notation of
Namikawa-Ueno.

***** The figure `[2I_0-m]`

in Namikawa-Ueno, page 159, must be denoted
by `[2I_0-(m+1)]`

.

The library syntax is `GEN `

.**genus2red**(GEN PQ, GEN p = NULL)

X being a non-singular hyperelliptic curve defined over a finite field, return the characteristic polynomial of the Frobenius automorphism. X can be given either by a squarefree polynomial P such that X: y^2 = P(x) or by a vector [P,Q] such that X: y^2 + Q(x) y = P(x) and Q^2+4 P is squarefree.

The library syntax is `GEN `

.**hyperellcharpoly**(GEN X)

Let X be the curve defined by y^2 = Q(x), where Q is a polynomial of degree d over ℚ and p ≥ d a prime such that X has good reduction at p return the matrix of the Frobenius endomorphism ϕ on the crystalline module D_p(X) = ℚ_p ⨂ H^1_{dR}(X/ℚ) with respect to the basis of the given model (ω, x ω,...,x^{g-1} ω), where ω = dx/(2 y) is the invariant differential, where g is the genus of X (either d = 2 g+1 or d = 2 g+2). The characteristic polynomial of ϕ is the numerator of the zeta-function of the reduction of the curve X modulo p. The matrix is computed to absolute p-adic precision p^n.

The library syntax is `GEN `

.**hyperellpadicfrobenius**(GEN Q, ulong p, long n)