ellminimaltwist

Function: ellminimaltwist
Section: elliptic_curves
C-Name: ellminimaltwist0
Prototype: GD0,L,
Help: ellminimaltwist(E, {flag=0}): E being 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.
Doc: 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 $\fl=0$, its minimal model has minimal
 discriminant, or if $\fl=1$, it has minimal conductor.

 In the example below, we find a curve with $j$-invariant $3$ and minimal
 conductor.
 \bprog
 ? E = ellminimalmodel(ellinit(ellfromj(3)));
 ? ellglobalred(E)[1]
 %2 = 357075
 ? D = ellminimaltwist(E,1)
 %3 = -15
 ? E2 = ellminimalmodel(elltwist(E,D));
 ? ellglobalred(E2)[1]
 %5 = 14283
 @eprog
 In the example below, $\fl=0$ and $\fl=1$ give different results.
 \bprog
 ? E = ellinit([1,0]);
 ? D0 = ellminimaltwist(E,0)
 %7 = 1
 ? D1 = ellminimaltwist(E,1)
 %8 = 8
 ? E0 = ellminimalmodel(elltwist(E,D0));
 ? [E0.disc, ellglobalred(E0)[1]]
 %10 = [-64, 64]
 ? E1 = ellminimalmodel(elltwist(E,D1));
 ? [E1.disc, ellglobalred(E1)[1]]
 %12 = [-4096, 32]
 @eprog

Variant: Also available are
 \fun{GEN}{ellminimaltwist}{E} for $\fl=0$, and
 \fun{GEN}{ellminimaltwistcond}{E} for $\fl=1$.