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

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

Bill:

Thanks for calling my attention to ell2cover.

In the past using the ?5 would pull up the elliptic curve commands available

now it is ?12 and a LOT more commands have been added.

It will take some time to become proficient in nearly all of them.

Randall

On 11/27/23 05:41, Bill Allombert wrote:
> 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