Bill Allombert on Sat, 02 Dec 2023 16:27:34 +0100

 Re: Interesting results using a NF for isogenous curves to determine possible regulator heights

• To: pari-users@pari.math.u-bordeaux.fr
• Subject: Re: Interesting results using a NF for isogenous curves to determine possible regulator heights
• From: Bill Allombert <Bill.Allombert@math.u-bordeaux.fr>
• Date: Sat, 2 Dec 2023 16:27:13 +0100
• Delivery-date: Sat, 02 Dec 2023 16:27:34 +0100
• Mail-followup-to: pari-users@pari.math.u-bordeaux.fr
• References: <ef62be0a-97c4-4333-af3f-cc1d222e6d2f@gmail.com>

```On Sat, Dec 02, 2023 at 04:29:40AM -0800, American Citizen wrote:
> Hello all:
>
> I used a number field NF=nfinit(a^2 - 5879094002709584654188067784965) for
> an elliptic curve (1)
>
> (1)   [1, 0, 0, -1473670506991667240419073314040,
> 487857899055649012085793427889299493072289600]
>
> which has 8 isogenous curves.
>
> Looking at the torsion points for (1) I selected x = 990529152804881 which
> gave the point p1 on the curve
>
> (2)  p1 = [990529152804881, Mod(1/2*a - 990529152804881/2, a^2 -
> 5879094002709584654188067784965)]
>
> Using the command: vector(8,i,ellheight(ellinit(k[i],NF),p1)) where k[1..8]
> is the isogenous curve for (1)  I was able

Make sure you need map p1 to k[i] first ! I do not think ellheight check
whether the point is actually on the curve...

Also note that we provide a function ellbsd:
The  object  E  being  an  elliptic  curve  over  a number field,  returns a
real number c such that the BSD conjecture predicts that
L_{E}^{(r)}(1)/r! = c R S where r is the rank, R the regulator and S the
cardinal of the Tate-Shafarevich group.

And even while you cannot compute  L_{E}^{(r)}(1)/r!, this value is the same
for all the isogenous curve, so you can find which ones has the smallest product
R*S

Cheers,
Bill

```