e = ellfromeqn(a*x^2+y^2 - (1+d*x^2*y^2))
e = ellfromj(3)
E = ellinit(e);
E.j \\ j-invariant
E.disc \\ discriminant

u = ffgen([101,2],’u);
E = ellinit([10,81*u+94],u);
[a1,a2,a3,a4,a6]=E[1..5]
E.e4
ellcard(E) \\ cardinal of E(F_q)
P = random(E) \\ random point on E(F_q)
Q = random(E) \\ another random point on E(F_q)
ellisoncurve(E, P) \\ check that the point is on the curve

elladd(E, P, Q) \\ P+Q in E
ellsub(E, P, Q) \\ P-Q in E, 
               set P=[0] to get the inverse of Q (or ellneg)
ellmul(E, P, 100) \\ 100.P in E
ellorder(E,P) \\order of P
[d1,d2]=ellgroup(E) \\ structure of E(F_q)

[G1,G2] = ellgenerators(E) \\ minimal generating set
ellorder(E,G1)
w = ellweilpairing(E,G1,G2,d1) \\ Weil pairing of G1 and G2
                                 of order d1
fforder(w)

e = random(d1);
S = ellmul(E,P,e) \\ e.P in E
elllog(E,S,P)
e

et = elltwist(E)
Et = ellinit(et);
ellap(E)
ellap(Et)

P3 = ellmul(E,G1,d1/3);
ellorder(E,P3)
[eq,iso] = ellisogeny(E,P3);
eq
iso
G1q = ellisogenyapply(iso, G1)
Eq = ellinit(eq); ellorder(Eq, G1q)

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

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

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

E=ellinit([0,1,1,-7,5]);
ellratpoints(E,100)
ellrank(E)
E = ellinit([-289,1]);
ellrank(E)

E = ellinit([-157^2,0]);
lfunorderzero(E)
P = ellheegner(E)

E=ellinit([0,1,1,-7,5]);
lfunorderzero(E)
P = ellheegner(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)

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

[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