Re: asking for a simple locally soluble algorithm for a quartic

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

On Sun, Nov 26, 2023 at 05:44:36PM -0800, American Citizen wrote:
> Does anyone have a simple GP-Pari script which outputs 0 for false and 1 for
> true when the input is a quartic in vector format: [a,b,c,d,e] where the
> quartic is a*x^4 + b*x^3 + c^x^2 + d^x +e and we are trying to find the
> everywhere_local_solubility of the quartic?

What do you really want to do ? All of this is implemented in PARI but
you have to pull the thread from the right end.
It seems like you are are trying to reimplement PARI piecewise in GP.

For example, you can do

P=Pol([a,b,c,d,e]) \\ convert [a,b,c,d,e] to a polynomial
hyperelldisc(P) \\ compute the discriminant of y^2-P

install(hyperell_locally_soluble,GG)
hyperell_locally_soluble(P,p) \\ check solubility of y^2-P at p

But if you have an elliptic curve E, ell2cover will return the list of
everywhere locally soluble 2-covers of E. 

Cheers,
Bill