Hi I haven't played with isogenies for a while but did so a lot recently. As I am no longer really familiar with the new structure in sage, I sent this message to this group in the hope that someone involved in isogenies in sage picks it up. I list a few separate issues and questions in one email. Sorry for its length.
I am happy to help (modulo time constraints) and feel free to contact me directly outside this forum. ------------------------------ a) Here is a first error, which I assume is a bug F.<s> = QuadraticField(-3) E = EllipticCurve(F,[0,0,1,0,0]) # has cm by O_F R.<x> = F[] phi = E.isogeny(x,codomain=E,degree=3) # is an associate to sqrt(-3) psi = 1 + phi psi.rational_maps() it causes boom with "TypeError: polynomial (=2) must be a polynomial" Related to this, one of the statements sage: (phi+1).degree(), (phi-1).degree() 7,7 is wrong as the only possible answers for associates of sqrt(-3) in F are 7,1 or 1,7 or 4,4. ------------------------------ b) Here is my next problem, continuing from the above phi7 = E.isogeny(x^3 + 3/14*s - 1/14, codomain=E, degree=7) xi = -2 + phi7 # should be the automorphism of order 3 xi.degree() goes boom with "ValueError: the two curves are not linked by a cyclic normalized isogeny of degree 7" at the last line not before. I know now that I should use .automorphism instead. ------------------------------ c) This is again a bug which results in an incorrect answer rather than an unexpected error. k.<z> = GF(25,"z") R.<x> = k[] A = EllipticCurve(k, [0,4,0,2,4]) f = x^3 + (3*z + 2)*x^2 + (z + 4)*x + z + 2 phi = A.isogeny(f) alpha = next(a for a in A.automorphisms() if a.order()==3) phi.kernel_polynomial(), (phi*alpha).kernel_polynomial(), (phi*alpha*alpha).kernel_polynomial() returns (x^3 + (3*z + 2)*x^2 + (z + 4)*x + z + 2, x^3 + 2*z*x^2 + 4*z*x + 4*z + 3, x^3 + (3*z + 2)*x^2 + (z + 4)*x + z + 2) while the first two are correct, the last is incorrect as it can definitely not be the same as the first. It should be f.subs(alpha.u^2*x+alpha.r) x^3 + 2*x + 4 ------------------------------ d) Here is an unexpected behaviour. An isogeny cannot be __call__ed on a point over a larger field, but it can be _eval-ed: This works fine: E = EllipticCurve(GF(7),[1,3]) # rather randomly chosen phi = E.isogenies_prime_degree(3)[0] # unique L.<z> = GF(7^2) EL = E.base_extend(L) P = EL([1,2+3*z]) # random phi._eval(P) but phi(P) throws "ValueError: 3*z + 2 is not in the image of (map internal to coercion system -- copy before use) Ring morphism: From: Finite Field of size 7 To: Finite Field in z of size 7^2" ------------------------------ e) Is there a way to base_extend phi to L? That shouldn't be hard to implement. My hand-on implementation of this is not really good, but it worked. Similarly, I wrote a "reduction" for an isogeny over a number field at a prime ideal where the model has good reduction. In both cases I extended/reduced the kernel polynomial and created a new isogeny on the extended/reduced curves and then adjusted it with an automorphism if the .scaling_factor did not extend/reduce to the new scaling_factor. Of course, for composite and sum morphisms I did it on the components. But it feels like one should be able to do this directly. Again, apologies for the length of this post. Chris -- 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/CABMU80cWGcb0Vm41Tumjjc5Aoh9mAZkwc_gBv52a9rSYNwT0Vg%40mail.gmail.com.
