This is probably naive. I'm also not sure how to easily consider P as an element of E(Kq). But perhaps we can compute the p-th division polynomial of P as an element of E(K). Embed this polynomial into Kq and find the x-coordinates by computing the roots in Kq. Then lift the roots in `E.change_ring(Kq)`.
This is how division_points works under the hood anyways but it won't require you to view P as an element of E(Kq). On Wednesday, February 5, 2020 at 3:01:06 PM UTC-8, Ahmed Matar wrote: > > I have an elliptic curve E defined over the rationals and K is an > imaginary quadratic field. I have a Heegner point P for E over K. I also > have a rational prime p. Let q be a prime of K above p. I would like to use > Sage to check whether the point P is divisible by p in E(K) and also in > E(Kq) where Kq is the completion of K at the prime q. To check this in E(K) > is easy; one can use the heegner_index() function or one can use the > division_points() function. I am wondering if there is a way in Sage to do > my required check in E(Kq). It seems to me that completions of number > fields at finite primes are not defined in Sage. One can define in sage Kq > as an extension field of Qp but then one must consider P as an element of > E(Kq) via an appropriate embedding and I'm unsure how to do this. > > Any ideas on how I can do my local computation? I need to do a bunch of > these local computations for a paper I'm working on. > -- You received this message because you are subscribed to the Google Groups "sage-nt" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To view this discussion on the web visit https://groups.google.com/d/msgid/sage-nt/c5c6e471-ca56-4670-8e1b-97c11bebc0a5%40googlegroups.com.
