Re: question on solving a certain cubic equation

[American Citizen <website.reader3@gmail.com>]

Bill

Thanks, I guess I overlooked multiplying the x^3 by the (-123980400)^2 
value and converting to a normal elliptic curve.

Randall

On 7/14/24 14:46, Bill Allombert wrote:
> On Sun, Jul 14, 2024 at 01:58:48PM -0700, American Citizen wrote:
> > Let the equation to be solved be
> >
> > (1)  y^2 = -12390400*x^3 + 5525390369*x^2 - 27063339058*x + 28033809489
> >
> > Is there any other way to find rational solutions for x, other than using
> > the hyperellratpoints() command?
> Sure, multiply both side by (-12390400)^2 and set
> Y = -12390400*y
> X = -12390400*x
>
> One get
> Y^2 = X^3+5525390369*X^2+335325596264243200*X+4303806841261381386240000
> and then we cab apply ellrank:
> %4 = [5,5,0,[[-1480827040,91525188474720],[-194902400,11892728848000],[-18295200,110732714400],[-11038720,1128797429760],[3908396800,381341478918400]]]
>
> So one get for example
> x = -1480827040/ -12390400
> y = 91525188474720/-12390400
>
> Cheers,
> Bill.