Re: Support for elliptic curves over number fields

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

On Sat, Jul 16, 2016 at 05:44:07PM +0100, John Cremona wrote:
> On 16 July 2016 at 17:06, Bill Allombert
> <Bill.Allombert@math.u-bordeaux1.fr> wrote:
> > Dear PARI developers,
> >
> > We have added basic support for elliptic curves over number fields and L
> > function of elliptic curves over number fields.
> 
> Excellent!  Some of this functionality is already mentioned on web
> pages such as http://www.lmfdb.org/EllipticCurve/3.3.148.1/356.1/a/1
> (after clicking the "Show commands for ... Pari/gp" near the top.  If
> those commands are not correct, please let me know (they were written
> by Aurel).  I notice that he did not include the field as a parameter
> in the ellinit().

Could you change the way the field is defined ?

Instead of 
  K = nfinit(x^3 - x^2 - 3*x + 1); a=x

  K = nfinit(a^3 - a^2 - 3*a + 1);
would be better

(Also LMFDB gives the L-function of E as
L(s,f)  = 1− 4^-s − 0.447·5^-s − 0.832·13^-s + 16^-s + 0.485·17^-s +...
but I do not think this is correct: it should be 
L(s,f)  = 1− 0.447·5^-s − 0.832·13^-s + 0.485·17^-s +...
The true L-function of E have a trivial Euler factor at 2.
I assume this is an instance of two L-functions differing by 
a single Euler factor at 2, which can happen in motivic weight 1).

Cheers,
Bill