Hi Chris, I wrote some of this stuff and will look into it, but there has been a lot of newer work by others, people interested only in finite base fields, and that might be relevant. But I'm away at the moment so I cannot do anything for a few days.
John On Thu, 8 Aug 2024, 14:25 Chris Wuthrich, <[email protected]> wrote: > 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 > <https://groups.google.com/d/msgid/sage-nt/CABMU80cWGcb0Vm41Tumjjc5Aoh9mAZkwc_gBv52a9rSYNwT0Vg%40mail.gmail.com?utm_medium=email&utm_source=footer> > . > -- 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/CAD0p0K6STKYPDNwvHPePd-foOfNjO-ZoR0KZodTDuMSi5sj%2Brg%40mail.gmail.com.
