Re: question on proving a certain elliptic curve has rank 0

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

On Tue, Dec 23, 2025 at 02:53:15PM -0800, American Citizen wrote:
> Hello:
> 
> According to GP-Pari the elliptic curve (1) has rank = 0.
> 
> (1)   y^2 = x^3 + 4*x^2 + 8*x + 8
> 
> This curve has torsion 2Z with the torsion point [-2,0]
> 
> How can I prove that the curve absolutely has rank = 0?

I am not sure what kind of proof you are aiming at.

? E=ellinit([0,4,0,8,8]);
? ellrank(E)
%4 = [0,0,0,[]]

so the rank is between 0 and 0.
(There is no solution everywhere locally).

(if you are concerned about the use ot the GRH, GRH is not used for curves
having a 2-torsion point).

Cheers,
Bill