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Bill Allombert on Sun, 14 Jul 2024 23:46:56 +0200
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Re: question on solving a certain cubic equation
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- To: pari-users@pari.math.u-bordeaux.fr
- Subject: Re: question on solving a certain cubic equation
- From: Bill Allombert <Bill.Allombert@math.u-bordeaux.fr>
- Date: Sun, 14 Jul 2024 23:46:49 +0200
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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.