Bill Allombert on Mon, 25 Jul 2016 11:05:08 +0200 |
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Re: Support for elliptic curves over number fields |
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