jvoight
commented
11 months ago Frank Calegari adds the following, concerning the mod 3 image in genus 2:
You mention having the degree 40 polynomial accessible (or the degree 40 number field). I’m not entirely sure how useful that is directly; certainly in my case it is very convenient to have both that polynomial and the degree 27 polynomial as well. Now Dave Roberts computed a degree 40 (and 80) polyomial in generic variables which makes computing the degree 40 polynomial instantaneous (if you have access to it!) but he doesn’t yet have the generic degree 27 polynomial. Also for me, the degree 40 (possibly reducible) polynomial was more useful than some field which may have degree less than 40 and would require special coding). One way to compute it would just be to note that the coefficients (starting with y^2 = x^6 + a x^4 + b x^3 + c x^2 + d x + e; note that the degree 27 polynomial is from PGSp_4 so twists make no difference so the x^6 coefficient can be 1) of this degree polynomial lie in Q[a,b,c,d,e] with some bounded degrees, so once one computes “enough” cases one can find the coefficients. On the other hand, this is for some computation which was particular to me…..
On the genus 2 pages, we list a defining polynomial for the image of the mod 2 Galois representation, and this is useful!
We should do this for elliptic curves over the rationals, as long as they are not too big to compute. Under the "Galois representations" section, we could do this for ell = 2,3,5,7, say, even if they are maximal--at least when the field is in the LMFDB?
If this was fun, we could also do it for elliptic curves over number fields.
Eventually, this could link in nicely to a section on mod ell Galois representations.