ellfromj

Function: ellfromj
Section: elliptic_curves
C-Name: ellfromj
Prototype: G
Help: ellfromj(j): returns the coefficients [a1,a2,a3,a4,a6] of a fixed
 elliptic curve with j-invariant j.
Doc: returns the coefficients $[a_1,a_2,a_3,a_4,a_6]$ of a fixed elliptic curve
 with $j$-invariant $j$. The given model is arbitrary; for instance, over the
 rationals, it is in general not minimal nor even integral.
 \bprog
 ? v = ellfromj(1/2)
 %1 = [0, 0, 0, 10365/4, 11937025/4]
 ? E = ellminimalmodel(ellinit(v)); E[1..5]
 %2 = [0, 0, 0, 41460, 190992400]
 ? F = ellminimalmodel(elltwist(E, 24)); F[1..5]
 %3 = [1, 0, 0, 72, 13822]
 ? [E.disc, F.disc]
 %4 = [-15763098924417024000, -82484842750]
 @eprog\noindent For rational $j$, the following program returns the integral
 curve of minimal discriminant and given $j$ invariant:
 \bprog
 ellfromjminimal(j)=
 { my(E = ellinit(ellfromj(j)));
   my(D = ellminimaltwist(E));

   ellminimalmodel(elltwist(E,D));
 }
 ? e = ellfromjminimal(1/2); e.disc
 %1 = -82484842750
 @eprog Using $\fl = 1$ in \kbd{ellminimaltwist} would instead return the
 curve of minimal conductor. For instance, if $j = 1728$, this would return a
 different curve (of conductor $32$ instead of $64$).