Re: Support for elliptic curves over number fields

[John Cremona <john.cremona@gmail.com>]

Thanks Bill -- I will change the nfinit line and will report the
L-function issue to the appropriate LMFDB-devs.

John

On 28 July 2016 at 11:04, Bill Allombert
<Bill.Allombert@math.u-bordeaux.fr> wrote:
> On Thu, Jul 28, 2016 at 12:40:12AM +0200, Bill Allombert wrote:
>> On Mon, Jul 25, 2016 at 11:05:06AM +0200, Bill Allombert wrote:
>> > 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).
>>
>> No actually the Euler factor at 89 is also different, and maybe I did
>> something wrong, but the LMFDB L-function does not satisfy the stated
>> functional equation.
>>
>> I used
>> L2=lfuncreate([H,0,[1/2,1/2,1/2,3/2,3/2,3/2],1,7797824,-1]);
>> where H is the list of Dirichlet coefficients)
>
> I just tried this one:
> <http://www.lmfdb.org/EllipticCurve/3.3.49.1/27.1/a/2>
> and both PARI and LMFDB agreed.
>
> So someone should double check
> <http://www.lmfdb.org/EllipticCurve/3.3.148.1/356.1/a/1>
>
> Cheers,
> Bill.
>