read("/where/you/saved/the/file/abgrp20200120.gp") \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ \\ Example: Ideal Class Group I K = bnfinit(y^2 + 2020); ClK = ag_clgp_of( K ); K.gen G = ag_G(ClK) type(ag_G(ClK)) K.cyc ag_DG(ClK) \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ \\ Example: Ideal Class Group II K = bnfinit(y^2 + 2020); ClK = ag_clgp_of( K ); G = ag_G(ClK) ag_id(ClK) ag_funmul(ClK) ag_mul(ClK, G[1,1], G[1,2]) elt = ag_mul(ClK, ag_pow(ClK, G[1,1], 23), ag_pow(ClK, G[1,2], -1)); ag_dlg(ClK, elt) \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ \\ Example: kernel of relative norm K0 = bnfinit(z^2 + 584*z + 27508); K0plus = bnrinit(K0, [1, [1, 1]], 1); ClK0plus = ag_clgp_of( K0plus ); KoverK0 = rnfinit(K0, y^2 - z); K = bnfinit(nfinit(KoverK0)); Km = bnrinit(K, m = 2, 1); ClKm = ag_clgp_of( Km ); ag_DG(ClKm) { ker = agmor_kernel( ClKm, ClK0plus, ((x)->rnfidealnormrel(KoverK0, idealhnf(K, x))) )}; ker[1] ker[2] == ClKm \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ \\ Example: SNF of subgroup K0 = bnfinit(z^2 + 584*z + 27508); K0plus = bnrinit(K0, [1, [1, 1]], 1); ClK0plus = ag_clgp_of( K0plus ); KoverK0 = rnfinit(K0, y^2 - z); K = bnfinit(nfinit(KoverK0)); Km = bnrinit(K, m = 2, 1); ClKm = ag_clgp_of( Km ); ag_DG(ClKm) { ker = agmor_kernel( ClKm, ClK0plus, ((x)->rnfidealnormrel(KoverK0, idealhnf(K, x))) )}; kersnf = fag_snfofsubgrp(ker); ag_DG(kersnf) ag_DG(ClKm) \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ \\ ATTENTION: K0 is different from the previous slide. \\ Example: $\calO_{K, 1 \mod \frakm}^\ast$ K0 = bnfinit(z^2 + 292*z + 14439); K0plus = bnrinit(K0, [1, [1, 1]], 1); ClK0plus = ag_clgp_of( K0plus ); KoverK0 = rnfinit(K0, y^2 - z); K = bnfinit(nfinit(KoverK0)); OKstar = ag_OKstar_of(K); OKmstar = ag_OKmstar_of(K, m = 2); H_OKm1star = agmor_kernel(OKstar, OKmstar, ((x)->x) )[1]; \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ \\ Example: $\frac{\calO_{K_0^+}^\times}{N_{K/K_0}\left(\calO_{K, 1 \mod m}^\times\right)}$ K0 = bnfinit(z^2 + 292*z + 14439); K0plus = bnrinit(K0, [1, [1, 1]], 1); ClK0plus = ag_clgp_of( K0plus ); KoverK0 = rnfinit(K0, y^2 - z); K = bnfinit(nfinit(KoverK0)); OK0star = ag_OKstar_of(K0); H_OK0plusstar = [2, 0; 0, 2]; H_relnorm_imageof_OK1star = [2, 0; 0, 6]; coker = ag_quogrp_oftwosubgroups(H_relnorm_imageof_OK1star, H_OK0plusstar, OK0star); \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\