E = EllipticCurve('681b'); E E.analytic_rank() E.sha_an() E.non_surjective() E.heegner_discriminants_list(10) n = E.heegner_index(-20); n parent(n) n^2 + 10 E = EllipticCurve('681b') L = E.Lseries_dokchitser() # T. Dokchitser L(1) L(2) L(1+I) E.Lseries_zeros(5) # M. Rubinstein E.Lseries_sympow(2,16) # M. Watkins E = EllipticCurve('37a') v = E.Lseries_values_along_line(1, 1+10*I, 300) w = [(z[1].real(), z[1].imag()) for z in v] L = line(w, rgbcolor=(0.5,0,0)) L.save('line.png') E = EllipticCurve('389a') L = E.Lseries_dokchitser() L(1+I) L(1+0.2*I, 50) E = EllipticCurve('37a') E.sha_an() E.non_surjective() E.sha_an() E.regulator() E.gens() E.heegner_discriminants(50) E.heegner_index(-7) # Kolyvagin ==> Sha trivial E.q_expansion(5) E.simon_two_descent () E.sea(next_prime(10^50)) M = ModularSymbols(Gamma1(13),2); M D = M.decomposition(3) [A.dimension() for A in D] save(M, 'modsym13') M = load('modsym13') D = M.decomposition(3) # instant! D[-1].hecke_operator(2).charpoly().factor() P. = ProjectiveSpace(3,QQ) C = P.subscheme([y^2-x*z, z^2-y*w, x*w-y*z]) len(C.irreducible_components()) # twisted cubic J = C.defining_ideal() G = J.groebner_fan() len(G.reduced_groebner_bases()) G.fvector() f = prod(J.gens()) # \/-- newton polytope NP = polymake.convex_hull(f.exponents()) NP.facets() G.interactive()