Re: question on recovering a single parameter from a minimal model elliptic curve

[Bill Allombert <Bill.Allombert@math.u-bordeaux.fr>]

On Mon, Feb 09, 2026 at 04:50:06PM -0800, American Citizen wrote:
> Hi:
> 
> I have an elliptic curve (not minimal model) defined for a rational number
> r.
> 
> > ? E1_min(r)[1..5]
> > %14 = [0, -1, 0, -274977905984, -55446464208117696]
> Given the minimal model E1_min coefficients, how can I recover r = 33/29?

Use the j-invariant, it is the same for E1 and E1_min.

nfroots(,numerator(E1(x).j - E1_min(33/29).j))
%412 = [-31/2,-33/29,-29/33,-2/31,2/31,29/33,33/29,31/2]~

Then you test whether (E.c6*Emin.c4)/(E.c4*Emin.c6) is a perfect square to
eleminate the wrong twists.

Cheers,
Bill.