Es=ellinit([a4,a6]);

El=ellinit([a1,a2,a3,a4,a6]);

Es.c4
Es.c6
Es.disc
Es.j

E = ellinit([0,1,1,-2,0]);
E.j
E.disc
N = ellglobalred(E)[1]
tor = elltors(E) \\ trivial

ellratpoints(E,10)
R = ellrank(E)
G = ellsaturation(E,R[4],1000)
elladd(E,G[1],G[2])
ellmul(E,G[1],3)

lfunparams(E)
lfunan(E,20)
lfunrootres(E)
ellrootno(E)
lfunorderzero(E)
lfun(E,1,2)
ellanalyticrank(E)

tam = elltamagawa(E)
bsd = E.omega[1]*tam
ellbsd(E)
reg = matdet(ellheightmatrix(E,G))
bsd * reg
lfun(E,1,2)/2!

E=ellinit(ellfromj(3)); E[1..5]
ellglobalred(E)[1]
E.disc
Em = ellminimalmodel(E); Em[1..5]
Em.disc

t = ellminimaltwist(E)
Et = elltwist(E,t); Et[1..5] 
Et = ellminimalmodel(Et); Et[1..5]
ellglobalred(Et)[1]
Et.disc

E=ellinit([0,1,1,-7,5]);
lfunorderzero(E)
P = ellheegner(E)
elltors(E)
ellisoncurve(E,P)
[L,M] = ellisomat(E);
M \\ isogeny matrix

[e2,iso2,isod2]=L[2]
E2 = ellinit(e2);
P2 = ellisogenyapply(iso2,P)
ellisoncurve(E2,P2)
ellheight(E2,P2)/ellheight(E,P)
Q=ellisogenyapply(isod2,P2)
ellmul(E,P,3)

E=ellinit("11a1");
ellglobalred(E)[1]
E=ellinit([3,4]);
ellidentify(E)
ellconvertname("1440i1")
ellsearch(27)

forell(e,1,10000,if(#e[3]==3,return(e)))
E3 = ellinit("5077a1");
ellgenerators(E3)
ellrank(E3)

E = ellinit("990h1");
[L,M] = ellweilcurve(E);
vector(#L,i,[ellidentify(L[i])[1][1],M[i]])

E  = ellinit("11a1");
E2 = ellinit("11a2");
E3 = ellinit("11a3");
Et = elltwist(E,-7);
E2t = elltwist(E2,-7);
E3t = elltwist(E3,-7);
F = intformal(Polrev(ellan(E,1000)));
h = (-13+sqrt(-7))/22;
z = 2*real(subst(F,x,exp(2*I*Pi*h))/sqrt(-7))

ellztopoint(Et,z)
ellztopoint(E2t,z)
ellztopoint(E3t,z)
ellztopoint(E3t,z*5)
ellmaninconstant(E)
ellmaninconstant(E2)
ellmaninconstant(E3)

E = ellinit("59097a4");
c = ellmaninconstant(E,1); \\ faster
d = ellmoddegree(E)
d*c^2
ellheight(E) \\ Falting height

nf  = nfinit(a^2-5);
phi = (1+a)/2;
E   = ellinit([1,phi+1,phi,phi,0],nf);
E.j
E.disc
N = ellglobalred(E)[1]
tor = elltors(E) \\ Z/8Z

idealhnf(E.nf,E.disc)
ellminimaldisc(E)
F = elltwist(E,5); lift(F[1..5])
idealhnf(F.nf,F.disc)
ellminimaldisc(F)
Fm = ellminimalmodel(ellinit(F,bnfinit(F.nf)));
lift(Fm[1..5])

[pr1, pr2] = idealprimedec(nf,31);
elllocalred(E,pr1) \\ multiplicative reduction
ellap(E,pr1) \\ -1: non-split
elllocalred(E,pr2) \\ good reduction
E2 = ellinit(E, pr2); \\ reduction of E mod pr2
E2.j
ellap(E2)
ellgroup(E2) \\ Z/24Z

per = E.omega[1][1]*E.omega[2][1];
tam = elltamagawa(E)
bsd = (per*tam) / (tor[1]^2*sqrt(abs(nf.disc)))
ellbsd(E)
L1 = lfun(E,1)
ellrootno(E)

ellisomat(E,,1)[2]

C = polclass(-23,,a)
E = ellinit(ellfromj(a), nfinit(C));
ellisomat(E)[2]
elliscm(E)

[P,j] = polcompositum(C, a^2+23, 1)[1]; 
E = ellinit(ellfromj(j), nfinit(P));
elliscm(E)
ellisomat(E)