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 assuming that r is at most the order of vanishing of the L-function at s = 1. (The result is wrong if r is strictly larger than the order of vanishing at 1.)

? e = ellinit("11a1"); \\ order of vanishing is 0 ? ellL1(e, 0) %2 = 0.2538418608559106843377589233 ? e = ellinit("389a1"); \\ order of vanishing is 2 ? ellL1(e, 0) %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 ? ellanalyticrank(ellinit([0, 0, 1, -7, 6])) time = 32 ms. %1 = [3, 10.3910994007158041] ? \p200 realprecision = 202 significant digits (200 digits displayed) ? ellL1(e, 3) time = 23,113 ms. %3 = 10.3910994007158041387518505103609170697263563756570092797 [...]

The library syntax is `GEN `

.**ellL1**(GEN e, long r, long prec)

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 = -3If 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. The curve must be given by an integral model, not necessarily minimal, although a minimal model will make the function faster.

The library syntax is `GEN `

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

, which
returns a **anellsmall**(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**(GEN e, GEN eps = NULL, 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 trace of Frobenius t for the elliptic curve E,
defined by the equation #E(**F**_q) = q+1 - t.

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) = sum_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

**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 q(p^{1/4}) using O(q^{1/4}) storage, hence becomes unreasonable when
q has about 30 digits. If the `seadata`

package is installed, the
`SEA`

algorithm becomes available, heuristically in Õ(log
q)^4, and primes of the order of 200 digits become feasible. In very small
characteristic (2,3,5,7 or 13), we use Harley's algorithm.

The library syntax is `GEN `

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

If z1 and z2 are points on the elliptic
curve E, assumed to be integral given by a minimal model, this function
computes the value of the canonical bilinear form on z1, z2:
( h(E,z1`+`

z2) - h(E,z1) - h(E,z2) ) / 2
where `+`

denotes of course addition on E. In addition, z1 or z2
(but not both) can be vectors or matrices.

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`

).

If the curve is defined over **Q**, p must be explicitly given and the
function computes the cardinal 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 associated 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 = (phi_n(P)psi_n(P) : omega_n(P) : psi_n(P)^3)
for some polynomials phi_n,omega_n,psi_n in
**Z**[a_1,a_2,a_3,a_4,a_6][X,Y]. We have f_n(X) = psi_n(X) for n odd, and
f_n(X) = psi_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 Pi/omega_2)^k
(1 + 2/zeta(1-k) sum_{n `>=`

0} n^{k-1}q^n / (1-q^n)),

where q = exp(2iPi tau) and tau := omega_1/omega_2 belongs to the
complex upper half-plane. It is also possible to directly input w =
[omega_1,omega_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 [eta_1,eta_2]
associated to the lattice basis *w* = [omega_1, omega_2].
Alternatively, *w* can be an elliptic curve E as output by
`ellinit`

, in which case, the quasi periods associated 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)

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 associated 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)

Calculates the arithmetic conductor, the global
minimal model of E and the global Tamagawa number c.
E must be an *ell* structure as output by `ellinit`

, defined over
**Q**. The result is a vector [N,v,c,F,L], where

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

***** v gives the coordinate change for E over **Q** to the minimal
integral model (see `ellminimalmodel`

),

***** 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 over **Z**.

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

,
where the local coordinate change has been deleted, and replaced by a 0.

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_1**Z**
x **Z**/d_2**Z**, 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 of the point z on the elliptic curve
E (defined over **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. *flag* selects
the algorithm used to compute the Archimedean local height. If *flag* = 0,
we use sigma and theta-functions and Silverman's trick (Computing
heights on elliptic curves, *Math. Comp.* **51**; note that
Silverman's height is twice ours). If
*flag* = 1, use Tate's 4^n algorithm. If *flag* = 2, use Mestre's AGM algorithm.
The latter converges quadratically and is much faster than the other two.

The library syntax is `GEN `

.
Also available is **ellheight0**(GEN E, GEN x, long flag, long prec)`GEN `

(**ghell**(GEN E, GEN x, long prec)*flag* = 2).

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 that this matrix should be divided by 2 to be in
accordance with certain normalizations. E is assumed to be integral,
given by a minimal model.

The library syntax is `GEN `

.**mathell**(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, associated 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.

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 != 5If 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,Delta,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)

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)

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 the prime p. E must be an *ell* structure as output by
`ellinit`

, and is assumed to have all its coefficients a_i in **Z**.
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+nu with nu > 0 means type I_nu; 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 [**Label: 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)

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)

Return the standard minimal integral model of the rational elliptic
curve E. If present, sets v to the corresponding change of variables,
which is a vector [u,r,s,t] with rational components. The return value is
identical to that of `ellchangecurve(E, v)`

.

The resulting model has integral coefficients, is everywhere minimal, a_1 is 0 or 1, a_2 is 0, 1 or -1 and a_3 is 0 or 1. Such a model is unique, and the vector v is unique if we specify that u is positive, which we do.

The library syntax is `GEN `

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

Return a vector [`eqn`

,t] where `eqn`

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 currently requires the package
`seadata`

to be installed and is limited to N < 500, N prime.

Let j be the j-invariant function. The polynomial `eqn`

satisfies
the following functional equation, which allows to compute the values of the
classical modular polynomial Phi_N of prime level N, such that
Phi_N(j(tau), j(Ntau)) = 0, while being much smaller than the latter:

***** for canonical type:
P(f(tau),j(tau)) = P(N^s/f(tau),j(N tau)) = 0,
where s = 12/gcd(12,N-1);

***** for Atkin type:
P(f(tau),j(tau)) = P(f(tau),j(N tau)) = 0.

In both cases, f is a suitable modular function (see below).

The following GP function returns values of the classical modular polynomial
by eliminating f(tau) in the above two equations, for N `<=`

31 or
N belongs to {41,47,59,71}.

classicaleqn(N, X='X, Y='Y)= { my(E=ellmodulareqn(N), P=E[1], t=E[2], Q, d); if(poldegree(P,'y)>2,error("level unavailable in classicaleqn")); if (t == 0, 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); , Q = subst(P,'y,Y); d = (X-Y)^(N+1)); polresultant(subst(P,'y,X), Q) / d; }

More precisely, let W_N(tau) = ({-1})/({N tau}) be the Atkin-Lehner involution; we have j(W_N(tau)) = j(N tau) and the function f also satisfies:

***** for canonical type:
f(W_N(tau)) = N^s/f(tau);

***** for Atkin type:
f(W_N(tau)) = f(tau).

Furthermore, for an equation of canonical type, f is the standard eta-quotient f(tau) = N^s (eta(N tau) / eta(tau) )^{2 s}, where eta is Dedekind's eta function, which is invariant under Gamma_0(N).

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)

Gives the order of the point z on the elliptic
curve E, defined over **Q** or a finite field.
If the curve is defined over **Q**, 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 ? E = ellinit(ellfromj(ffgen(5^10))); ? ellcard(E) %5 = 9762580 ? P = random(E); ellorder(E, P) %6 = 4881290 ? p = 2^160+7; E = ellinit([1,2], p); ? N = ellcard(E) %8 = 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 [**Label: 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)

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 tau := 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], [eta_1,eta_2]], where
eta_1 and eta_2 are the quasi-periods associated to
[W_1,W_2], satisfying eta_1 W_2 - eta_2 W_1 = 2 i Pi.

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**/Lambda is a complex elliptic curve (Lambda =
`E.omega`

),
computes a complex number z, well-defined modulo the lattice Lambda,
corresponding to the point P; i.e. such that
P = [wp_Lambda(z),wp'_Lambda(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 belongs to E(**R**), we have more precisely, 0 \leq
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/\bar{**Q**_p} is analytically
isomorphic to \bar{**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 phi: E(**Q**_p) ~ **Q**_p^*/q^**Z** and the function returns
phi(P) belongs to **Q**_p.

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

is a
`t_POLMOD`

), we only have an isomorphism phi: E(K) ~ K^*/q^**Z** over
the unramified quadratic extension K/**Q**_p. In this case, the output
phi(P) belongs to 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 \idx{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)

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 condutor, 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 is a full curve name, e.g. `"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 \idx{Mordell-Weil
group} associated 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 sigma 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.
sigma(z, L) = z prod_{omega belongs to L^*} (1 -
(z)/(omega))e^{(z)/(omega) + (z^2)/(2omega^2)}.
It is also possible to directly input L = [omega_1,omega_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(sigma(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 Phi: X_0(N)\to E
be a modular parametrization; the pullback by Phi of the
Néron differential du/(2v+a_1u+a_3) is equal to 2iPi
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(2iPi z).

The algorithm assumes that E is a *strong* Weil curve
and that the Manin constant is equal to 1: in fact, f(x) = sum_{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 Q*, 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 `ellinit`

, defined over

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

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

If *flag* = 0, find rational roots of division polynomials.

If *flag* = 1, use Lutz-Nagell (*much* slower).

If *flag* = 2, use Doud's algorithm: bound torsion by computing #E(**F**_p)
for small primes of good reduction, then look for torsion points using
Weierstrass wp function (and Mazur's classification). For this variant,
E must be an *ell*.

The library syntax is `GEN `

.
Also available is **elltors0**(GEN E, long flag)`GEN `

for **elltors**(GEN E)`elltors(E, 0)`

.

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 wp function attached to
the lattice w as given by `ellperiods`

. It is also possible to
directly input w = [omega_1,omega_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 = 3.9413112427016474646048282462709151389One 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 wp(z), 1: compute
[wp(z),wp'(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)

Computes the value at z of the Weierstrass zeta 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.
zeta(z, L) = (1)/(z) + z^2sum_{omega belongs to L^*}
(1)/(omega^2(z-omega)).
It is also possible to directly input w = [omega_1,omega_2],
or an elliptic curve E as given by `ellinit`

(w = `E.omega`

).
The quasi-periods of zeta, such that
zeta(z + aomega_1 + bomega_2) = zeta(z) + aeta_1 + beta_2
for integers a and b are obtained as eta_i = 2zeta(omega_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 = 0.84721308479397908660649912348219163647One 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
wp-function and its derivative. More
precisely, we have
x = wp(z) - b_2/12, y = wp'(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 Q,P be polynomials with integer coefficients.
Determines the reduction at p > 2 of the (proper, smooth) genus 2
curve C/**Q**, defined by the hyperelliptic equation y^2+Qy = P. (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 rewritten from an implementation of Liu's algorithm by
Cohen and Liu (1994), `genus2reduction-0.3`

, see
`http://www.math.u-bordeaux1.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(0,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 are 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 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, \kbd{[I{
2}-II-5]} means `[I_2-II-5]`

.

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

and `[K'-K-m]`

are the same.

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

is `[K'-K-alpha]`

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 Q, GEN P, GEN p = NULL)