# Abelian variety isogeny classes downloaded from the LMFDB on 01 July 2026. # Search link: https://www.lmfdb.org/Variety/Abelian/Fq/3/2/ # Query "{'q': 2, 'g': 3}" returned 215 classes, sorted by dimension. # Each entry in the following data list has the form: # [Label, Dimension, Base field, L-polynomial, $p$-rank, Isogeny factors] # For more details, see the definitions at the bottom of the file. "3.2.ag_s_abg" 3 2 [1, -6, 18, -32, 36, -24, 8] 0 [["1.2.ac", 3]] "3.2.af_n_aw" 3 2 [1, -5, 13, -22, 26, -20, 8] 2 [["1.2.ac", 1], ["2.2.ad_f", 1]] "3.2.af_o_ay" 3 2 [1, -5, 14, -24, 28, -20, 8] 1 [["1.2.ac", 2], ["1.2.ab", 1]] "3.2.ae_i_am" 3 2 [1, -4, 8, -12, 16, -16, 8] 0 [["1.2.ac", 1], ["2.2.ac_c", 1]] "3.2.ae_j_ap" 3 2 [1, -4, 9, -15, 18, -16, 8] 3 [["3.2.ae_j_ap", 1]] "3.2.ae_j_ao" 3 2 [1, -4, 9, -14, 18, -16, 8] 2 [["1.2.ac", 1], ["2.2.ac_d", 1]] "3.2.ae_k_ar" 3 2 [1, -4, 10, -17, 20, -16, 8] 3 [["1.2.ab", 1], ["2.2.ad_f", 1]] "3.2.ae_k_aq" 3 2 [1, -4, 10, -16, 20, -16, 8] 0 [["1.2.ac", 2], ["1.2.a", 1]] "3.2.ae_l_as" 3 2 [1, -4, 11, -18, 22, -16, 8] 2 [["1.2.ac", 1], ["1.2.ab", 2]] "3.2.ad_c_b" 3 2 [1, -3, 2, 1, 4, -12, 8] 3 [["3.2.ad_c_b", 1]] "3.2.ad_d_ac" 3 2 [1, -3, 3, -2, 6, -12, 8] 2 [["1.2.ac", 1], ["2.2.ab_ab", 1]] "3.2.ad_e_ae" 3 2 [1, -3, 4, -4, 8, -12, 8] 1 [["1.2.ac", 1], ["2.2.ab_a", 1]] "3.2.ad_f_ah" 3 2 [1, -3, 5, -7, 10, -12, 8] 3 [["3.2.ad_f_ah", 1]] "3.2.ad_f_ag" 3 2 [1, -3, 5, -6, 10, -12, 8] 2 [["1.2.ac", 1], ["2.2.ab_b", 1]] "3.2.ad_g_ak" 3 2 [1, -3, 6, -10, 12, -12, 8] 1 [["1.2.ab", 1], ["2.2.ac_c", 1]] "3.2.ad_g_aj" 3 2 [1, -3, 6, -9, 12, -12, 8] 3 [["3.2.ad_g_aj", 1]] "3.2.ad_g_ai" 3 2 [1, -3, 6, -8, 12, -12, 8] 1 [["1.2.ac", 2], ["1.2.b", 1]] "3.2.ad_h_am" 3 2 [1, -3, 7, -12, 14, -12, 8] 2 [["1.2.a", 1], ["2.2.ad_f", 1]] "3.2.ad_h_al" 3 2 [1, -3, 7, -11, 14, -12, 8] 3 [["1.2.ab", 1], ["2.2.ac_d", 1]] "3.2.ad_h_ak" 3 2 [1, -3, 7, -10, 14, -12, 8] 2 [["1.2.ac", 1], ["2.2.ab_d", 1]] "3.2.ad_i_am" 3 2 [1, -3, 8, -12, 16, -12, 8] 1 [["1.2.ac", 1], ["1.2.ab", 1], ["1.2.a", 1]] "3.2.ad_j_an" 3 2 [1, -3, 9, -13, 18, -12, 8] 3 [["1.2.ab", 3]] "3.2.ac_ac_i" 3 2 [1, -2, -2, 8, -4, -8, 8] 0 [["1.2.ac", 1], ["2.2.a_ae", 1]] "3.2.ac_ab_g" 3 2 [1, -2, -1, 6, -2, -8, 8] 2 [["1.2.ac", 1], ["2.2.a_ad", 1]] "3.2.ac_a_d" 3 2 [1, -2, 0, 3, 0, -8, 8] 3 [["3.2.ac_a_d", 1]] "3.2.ac_a_e" 3 2 [1, -2, 0, 4, 0, -8, 8] 0 [["1.2.ac", 1], ["2.2.a_ac", 1]] "3.2.ac_b_a" 3 2 [1, -2, 1, 0, 2, -8, 8] 2 [["3.2.ac_b_a", 1]] "3.2.ac_b_b" 3 2 [1, -2, 1, 1, 2, -8, 8] 3 [["3.2.ac_b_b", 1]] "3.2.ac_b_c" 3 2 [1, -2, 1, 2, 2, -8, 8] 2 [["1.2.ac", 1], ["2.2.a_ab", 1]] "3.2.ac_c_ad" 3 2 [1, -2, 2, -3, 4, -8, 8] 3 [["1.2.ab", 1], ["2.2.ab_ab", 1]] "3.2.ac_c_ac" 3 2 [1, -2, 2, -2, 4, -8, 8] 0 [["3.2.ac_c_ac", 1]] "3.2.ac_c_ab" 3 2 [1, -2, 2, -1, 4, -8, 8] 3 [["3.2.ac_c_ab", 1]] "3.2.ac_c_a" 3 2 [1, -2, 2, 0, 4, -8, 8] 0 [["1.2.ac", 2], ["1.2.c", 1]] "3.2.ac_d_ag" 3 2 [1, -2, 3, -6, 6, -8, 8] 2 [["3.2.ac_d_ag", 1]] "3.2.ac_d_af" 3 2 [1, -2, 3, -5, 6, -8, 8] 3 [["3.2.ac_d_af", 1]] "3.2.ac_d_ae" 3 2 [1, -2, 3, -4, 6, -8, 8] 2 [["1.2.ab", 1], ["2.2.ab_a", 1]] "3.2.ac_d_ad" 3 2 [1, -2, 3, -3, 6, -8, 8] 3 [["3.2.ac_d_ad", 1]] "3.2.ac_d_ac" 3 2 [1, -2, 3, -2, 6, -8, 8] 2 [["1.2.ac", 1], ["2.2.a_b", 1]] "3.2.ac_e_ai" 3 2 [1, -2, 4, -8, 8, -8, 8] 0 [["1.2.a", 1], ["2.2.ac_c", 1]] "3.2.ac_e_ah" 3 2 [1, -2, 4, -7, 8, -8, 8] 3 [["1.2.b", 1], ["2.2.ad_f", 1]] "3.2.ac_e_ag" 3 2 [1, -2, 4, -6, 8, -8, 8] 0 [["3.2.ac_e_ag", 1]] "3.2.ac_e_af" 3 2 [1, -2, 4, -5, 8, -8, 8] 3 [["1.2.ab", 1], ["2.2.ab_b", 1]] "3.2.ac_e_ae" 3 2 [1, -2, 4, -4, 8, -8, 8] 0 [["1.2.ac", 1], ["2.2.a_c", 1]] "3.2.ac_f_ai" 3 2 [1, -2, 5, -8, 10, -8, 8] 2 [["1.2.a", 1], ["2.2.ac_d", 1]] "3.2.ac_f_ah" 3 2 [1, -2, 5, -7, 10, -8, 8] 3 [["3.2.ac_f_ah", 1]] "3.2.ac_f_ag" 3 2 [1, -2, 5, -6, 10, -8, 8] 2 [["1.2.ac", 1], ["1.2.ab", 1], ["1.2.b", 1]] "3.2.ac_g_ai" 3 2 [1, -2, 6, -8, 12, -8, 8] 0 [["1.2.ac", 1], ["1.2.a", 2]] "3.2.ac_g_ah" 3 2 [1, -2, 6, -7, 12, -8, 8] 3 [["1.2.ab", 1], ["2.2.ab_d", 1]] "3.2.ac_h_ai" 3 2 [1, -2, 7, -8, 14, -8, 8] 2 [["1.2.ab", 2], ["1.2.a", 1]] "3.2.ab_ac_e" 3 2 [1, -1, -2, 4, -4, -4, 8] 1 [["1.2.ab", 1], ["2.2.a_ae", 1]] "3.2.ab_ab_c" 3 2 [1, -1, -1, 2, -2, -4, 8] 2 [["3.2.ab_ab_c", 1]] "3.2.ab_ab_d" 3 2 [1, -1, -1, 3, -2, -4, 8] 3 [["1.2.ab", 1], ["2.2.a_ad", 1]] "3.2.ab_ab_e" 3 2 [1, -1, -1, 4, -2, -4, 8] 2 [["3.2.ab_ab_e", 1]] "3.2.ab_ab_f" 3 2 [1, -1, -1, 5, -2, -4, 8] 3 [["3.2.ab_ab_f", 1]] "3.2.ab_ab_g" 3 2 [1, -1, -1, 6, -2, -4, 8] 2 [["1.2.ac", 1], ["2.2.b_ab", 1]] "3.2.ab_a_ab" 3 2 [1, -1, 0, -1, 0, -4, 8] 3 [["3.2.ab_a_ab", 1]] "3.2.ab_a_a" 3 2 [1, -1, 0, 0, 0, -4, 8] 1 [["3.2.ab_a_a", 1]] "3.2.ab_a_b" 3 2 [1, -1, 0, 1, 0, -4, 8] 3 [["3.2.ab_a_b", 1]] "3.2.ab_a_c" 3 2 [1, -1, 0, 2, 0, -4, 8] 1 [["1.2.ab", 1], ["2.2.a_ac", 1]] "3.2.ab_a_d" 3 2 [1, -1, 0, 3, 0, -4, 8] 3 [["3.2.ab_a_d", 1]] "3.2.ab_a_e" 3 2 [1, -1, 0, 4, 0, -4, 8] 1 [["1.2.ac", 1], ["2.2.b_a", 1]] "3.2.ab_b_ae" 3 2 [1, -1, 1, -4, 2, -4, 8] 2 [["1.2.a", 1], ["2.2.ab_ab", 1]] "3.2.ab_b_ad" 3 2 [1, -1, 1, -3, 2, -4, 8] 3 [["3.2.ab_b_ad", 1]] "3.2.ab_b_ac" 3 2 [1, -1, 1, -2, 2, -4, 8] 2 [["1.2.c", 1], ["2.2.ad_f", 1]] "3.2.ab_b_ab" 3 2 [1, -1, 1, -1, 2, -4, 8] 3 [["3.2.ab_b_ab", 1]] "3.2.ab_b_a" 3 2 [1, -1, 1, 0, 2, -4, 8] 2 [["3.2.ab_b_a", 1]] "3.2.ab_b_b" 3 2 [1, -1, 1, 1, 2, -4, 8] 3 [["1.2.ab", 1], ["2.2.a_ab", 1]] "3.2.ab_b_c" 3 2 [1, -1, 1, 2, 2, -4, 8] 2 [["1.2.ac", 1], ["2.2.b_b", 1]] "3.2.ab_c_ag" 3 2 [1, -1, 2, -6, 4, -4, 8] 1 [["1.2.b", 1], ["2.2.ac_c", 1]] "3.2.ab_c_af" 3 2 [1, -1, 2, -5, 4, -4, 8] 3 [["3.2.ab_c_af", 1]] "3.2.ab_c_ae" 3 2 [1, -1, 2, -4, 4, -4, 8] 1 [["1.2.a", 1], ["2.2.ab_a", 1]] "3.2.ab_c_ad" 3 2 [1, -1, 2, -3, 4, -4, 8] 3 [["3.2.ab_c_ad", 1]] "3.2.ab_c_ac" 3 2 [1, -1, 2, -2, 4, -4, 8] 1 [["3.2.ab_c_ac", 1]] "3.2.ab_c_ab" 3 2 [1, -1, 2, -1, 4, -4, 8] 3 [["3.2.ab_c_ab", 1]] "3.2.ab_c_a" 3 2 [1, -1, 2, 0, 4, -4, 8] 1 [["1.2.ac", 1], ["1.2.ab", 1], ["1.2.c", 1]] "3.2.ab_d_af" 3 2 [1, -1, 3, -5, 6, -4, 8] 3 [["1.2.b", 1], ["2.2.ac_d", 1]] "3.2.ab_d_ae" 3 2 [1, -1, 3, -4, 6, -4, 8] 2 [["1.2.a", 1], ["2.2.ab_b", 1]] "3.2.ab_d_ad" 3 2 [1, -1, 3, -3, 6, -4, 8] 3 [["3.2.ab_d_ad", 1]] "3.2.ab_d_ac" 3 2 [1, -1, 3, -2, 6, -4, 8] 2 [["1.2.ac", 1], ["2.2.b_d", 1]] "3.2.ab_d_ab" 3 2 [1, -1, 3, -1, 6, -4, 8] 3 [["1.2.ab", 1], ["2.2.a_b", 1]] "3.2.ab_e_ae" 3 2 [1, -1, 4, -4, 8, -4, 8] 1 [["1.2.ac", 1], ["1.2.a", 1], ["1.2.b", 1]] "3.2.ab_e_ad" 3 2 [1, -1, 4, -3, 8, -4, 8] 3 [["3.2.ab_e_ad", 1]] "3.2.ab_e_ac" 3 2 [1, -1, 4, -2, 8, -4, 8] 1 [["1.2.ab", 1], ["2.2.a_c", 1]] "3.2.ab_f_ae" 3 2 [1, -1, 5, -4, 10, -4, 8] 2 [["1.2.a", 1], ["2.2.ab_d", 1]] "3.2.ab_f_ad" 3 2 [1, -1, 5, -3, 10, -4, 8] 3 [["1.2.ab", 2], ["1.2.b", 1]] "3.2.ab_g_ae" 3 2 [1, -1, 6, -4, 12, -4, 8] 1 [["1.2.ab", 1], ["1.2.a", 2]] "3.2.a_ac_a" 3 2 [1, 0, -2, 0, -4, 0, 8] 0 [["1.2.a", 1], ["2.2.a_ae", 1]] "3.2.a_ab_ac" 3 2 [1, 0, -1, -2, -2, 0, 8] 2 [["3.2.a_ab_ac", 1]] "3.2.a_ab_ab" 3 2 [1, 0, -1, -1, -2, 0, 8] 3 [["3.2.a_ab_ab", 1]] "3.2.a_ab_a" 3 2 [1, 0, -1, 0, -2, 0, 8] 2 [["1.2.a", 1], ["2.2.a_ad", 1]] "3.2.a_ab_b" 3 2 [1, 0, -1, 1, -2, 0, 8] 3 [["3.2.a_ab_b", 1]] "3.2.a_ab_c" 3 2 [1, 0, -1, 2, -2, 0, 8] 2 [["3.2.a_ab_c", 1]] "3.2.a_a_af" 3 2 [1, 0, 0, -5, 0, 0, 8] 3 [["1.2.b", 1], ["2.2.ab_ab", 1]] "3.2.a_a_ae" 3 2 [1, 0, 0, -4, 0, 0, 8] 0 [["1.2.c", 1], ["2.2.ac_c", 1]] "3.2.a_a_ad" 3 2 [1, 0, 0, -3, 0, 0, 8] 3 [["3.2.a_a_ad", 1]] "3.2.a_a_ac" 3 2 [1, 0, 0, -2, 0, 0, 8] 0 [["3.2.a_a_ac", 1]] "3.2.a_a_ab" 3 2 [1, 0, 0, -1, 0, 0, 8] 3 [["3.2.a_a_ab", 1]] "3.2.a_a_a" 3 2 [1, 0, 0, 0, 0, 0, 8] 0 [["1.2.a", 1], ["2.2.a_ac", 1]] "3.2.a_a_b" 3 2 [1, 0, 0, 1, 0, 0, 8] 3 [["3.2.a_a_b", 1]] "3.2.a_a_c" 3 2 [1, 0, 0, 2, 0, 0, 8] 0 [["3.2.a_a_c", 1]] "3.2.a_a_d" 3 2 [1, 0, 0, 3, 0, 0, 8] 3 [["3.2.a_a_d", 1]] "3.2.a_a_e" 3 2 [1, 0, 0, 4, 0, 0, 8] 0 [["1.2.ac", 1], ["2.2.c_c", 1]] "3.2.a_a_f" 3 2 [1, 0, 0, 5, 0, 0, 8] 3 [["1.2.ab", 1], ["2.2.b_ab", 1]] "3.2.a_b_ae" 3 2 [1, 0, 1, -4, 2, 0, 8] 2 [["1.2.b", 1], ["2.2.ab_a", 1]] "3.2.a_b_ad" 3 2 [1, 0, 1, -3, 2, 0, 8] 3 [["3.2.a_b_ad", 1]] "3.2.a_b_ac" 3 2 [1, 0, 1, -2, 2, 0, 8] 2 [["1.2.c", 1], ["2.2.ac_d", 1]] "3.2.a_b_ab" 3 2 [1, 0, 1, -1, 2, 0, 8] 3 [["3.2.a_b_ab", 1]] "3.2.a_b_a" 3 2 [1, 0, 1, 0, 2, 0, 8] 2 [["1.2.a", 1], ["2.2.a_ab", 1]] "3.2.a_b_b" 3 2 [1, 0, 1, 1, 2, 0, 8] 3 [["3.2.a_b_b", 1]] "3.2.a_b_c" 3 2 [1, 0, 1, 2, 2, 0, 8] 2 [["1.2.ac", 1], ["2.2.c_d", 1]] "3.2.a_b_d" 3 2 [1, 0, 1, 3, 2, 0, 8] 3 [["3.2.a_b_d", 1]] "3.2.a_b_e" 3 2 [1, 0, 1, 4, 2, 0, 8] 2 [["1.2.ab", 1], ["2.2.b_a", 1]] "3.2.a_c_ad" 3 2 [1, 0, 2, -3, 4, 0, 8] 3 [["1.2.b", 1], ["2.2.ab_b", 1]] "3.2.a_c_ac" 3 2 [1, 0, 2, -2, 4, 0, 8] 0 [["3.2.a_c_ac", 1]] "3.2.a_c_ab" 3 2 [1, 0, 2, -1, 4, 0, 8] 3 [["3.2.a_c_ab", 1]] "3.2.a_c_a" 3 2 [1, 0, 2, 0, 4, 0, 8] 0 [["1.2.ac", 1], ["1.2.a", 1], ["1.2.c", 1]] "3.2.a_c_b" 3 2 [1, 0, 2, 1, 4, 0, 8] 3 [["3.2.a_c_b", 1]] "3.2.a_c_c" 3 2 [1, 0, 2, 2, 4, 0, 8] 0 [["3.2.a_c_c", 1]] "3.2.a_c_d" 3 2 [1, 0, 2, 3, 4, 0, 8] 3 [["1.2.ab", 1], ["2.2.b_b", 1]] "3.2.a_d_ac" 3 2 [1, 0, 3, -2, 6, 0, 8] 2 [["1.2.ac", 1], ["1.2.b", 2]] "3.2.a_d_ab" 3 2 [1, 0, 3, -1, 6, 0, 8] 3 [["3.2.a_d_ab", 1]] "3.2.a_d_a" 3 2 [1, 0, 3, 0, 6, 0, 8] 2 [["1.2.a", 1], ["2.2.a_b", 1]] "3.2.a_d_b" 3 2 [1, 0, 3, 1, 6, 0, 8] 3 [["3.2.a_d_b", 1]] "3.2.a_d_c" 3 2 [1, 0, 3, 2, 6, 0, 8] 2 [["1.2.ab", 2], ["1.2.c", 1]] "3.2.a_e_ab" 3 2 [1, 0, 4, -1, 8, 0, 8] 3 [["1.2.b", 1], ["2.2.ab_d", 1]] "3.2.a_e_a" 3 2 [1, 0, 4, 0, 8, 0, 8] 0 [["1.2.a", 1], ["2.2.a_c", 1]] "3.2.a_e_b" 3 2 [1, 0, 4, 1, 8, 0, 8] 3 [["1.2.ab", 1], ["2.2.b_d", 1]] "3.2.a_f_a" 3 2 [1, 0, 5, 0, 10, 0, 8] 2 [["1.2.ab", 1], ["1.2.a", 1], ["1.2.b", 1]] "3.2.a_g_a" 3 2 [1, 0, 6, 0, 12, 0, 8] 0 [["1.2.a", 3]] "3.2.b_ac_ae" 3 2 [1, 1, -2, -4, -4, 4, 8] 1 [["1.2.b", 1], ["2.2.a_ae", 1]] "3.2.b_ab_ag" 3 2 [1, 1, -1, -6, -2, 4, 8] 2 [["1.2.c", 1], ["2.2.ab_ab", 1]] "3.2.b_ab_af" 3 2 [1, 1, -1, -5, -2, 4, 8] 3 [["3.2.b_ab_af", 1]] "3.2.b_ab_ae" 3 2 [1, 1, -1, -4, -2, 4, 8] 2 [["3.2.b_ab_ae", 1]] "3.2.b_ab_ad" 3 2 [1, 1, -1, -3, -2, 4, 8] 3 [["1.2.b", 1], ["2.2.a_ad", 1]] "3.2.b_ab_ac" 3 2 [1, 1, -1, -2, -2, 4, 8] 2 [["3.2.b_ab_ac", 1]] "3.2.b_a_ae" 3 2 [1, 1, 0, -4, 0, 4, 8] 1 [["1.2.c", 1], ["2.2.ab_a", 1]] "3.2.b_a_ad" 3 2 [1, 1, 0, -3, 0, 4, 8] 3 [["3.2.b_a_ad", 1]] "3.2.b_a_ac" 3 2 [1, 1, 0, -2, 0, 4, 8] 1 [["1.2.b", 1], ["2.2.a_ac", 1]] "3.2.b_a_ab" 3 2 [1, 1, 0, -1, 0, 4, 8] 3 [["3.2.b_a_ab", 1]] "3.2.b_a_a" 3 2 [1, 1, 0, 0, 0, 4, 8] 1 [["3.2.b_a_a", 1]] "3.2.b_a_b" 3 2 [1, 1, 0, 1, 0, 4, 8] 3 [["3.2.b_a_b", 1]] "3.2.b_b_ac" 3 2 [1, 1, 1, -2, 2, 4, 8] 2 [["1.2.c", 1], ["2.2.ab_b", 1]] "3.2.b_b_ab" 3 2 [1, 1, 1, -1, 2, 4, 8] 3 [["1.2.b", 1], ["2.2.a_ab", 1]] "3.2.b_b_a" 3 2 [1, 1, 1, 0, 2, 4, 8] 2 [["3.2.b_b_a", 1]] "3.2.b_b_b" 3 2 [1, 1, 1, 1, 2, 4, 8] 3 [["3.2.b_b_b", 1]] "3.2.b_b_c" 3 2 [1, 1, 1, 2, 2, 4, 8] 2 [["1.2.ac", 1], ["2.2.d_f", 1]] "3.2.b_b_d" 3 2 [1, 1, 1, 3, 2, 4, 8] 3 [["3.2.b_b_d", 1]] "3.2.b_b_e" 3 2 [1, 1, 1, 4, 2, 4, 8] 2 [["1.2.a", 1], ["2.2.b_ab", 1]] "3.2.b_c_a" 3 2 [1, 1, 2, 0, 4, 4, 8] 1 [["1.2.ac", 1], ["1.2.b", 1], ["1.2.c", 1]] "3.2.b_c_b" 3 2 [1, 1, 2, 1, 4, 4, 8] 3 [["3.2.b_c_b", 1]] "3.2.b_c_c" 3 2 [1, 1, 2, 2, 4, 4, 8] 1 [["3.2.b_c_c", 1]] "3.2.b_c_d" 3 2 [1, 1, 2, 3, 4, 4, 8] 3 [["3.2.b_c_d", 1]] "3.2.b_c_e" 3 2 [1, 1, 2, 4, 4, 4, 8] 1 [["1.2.a", 1], ["2.2.b_a", 1]] "3.2.b_c_f" 3 2 [1, 1, 2, 5, 4, 4, 8] 3 [["3.2.b_c_f", 1]] "3.2.b_c_g" 3 2 [1, 1, 2, 6, 4, 4, 8] 1 [["1.2.ab", 1], ["2.2.c_c", 1]] "3.2.b_d_b" 3 2 [1, 1, 3, 1, 6, 4, 8] 3 [["1.2.b", 1], ["2.2.a_b", 1]] "3.2.b_d_c" 3 2 [1, 1, 3, 2, 6, 4, 8] 2 [["1.2.c", 1], ["2.2.ab_d", 1]] "3.2.b_d_d" 3 2 [1, 1, 3, 3, 6, 4, 8] 3 [["3.2.b_d_d", 1]] "3.2.b_d_e" 3 2 [1, 1, 3, 4, 6, 4, 8] 2 [["1.2.a", 1], ["2.2.b_b", 1]] "3.2.b_d_f" 3 2 [1, 1, 3, 5, 6, 4, 8] 3 [["1.2.ab", 1], ["2.2.c_d", 1]] "3.2.b_e_c" 3 2 [1, 1, 4, 2, 8, 4, 8] 1 [["1.2.b", 1], ["2.2.a_c", 1]] "3.2.b_e_d" 3 2 [1, 1, 4, 3, 8, 4, 8] 3 [["3.2.b_e_d", 1]] "3.2.b_e_e" 3 2 [1, 1, 4, 4, 8, 4, 8] 1 [["1.2.ab", 1], ["1.2.a", 1], ["1.2.c", 1]] "3.2.b_f_d" 3 2 [1, 1, 5, 3, 10, 4, 8] 3 [["1.2.ab", 1], ["1.2.b", 2]] "3.2.b_f_e" 3 2 [1, 1, 5, 4, 10, 4, 8] 2 [["1.2.a", 1], ["2.2.b_d", 1]] "3.2.b_g_e" 3 2 [1, 1, 6, 4, 12, 4, 8] 1 [["1.2.a", 2], ["1.2.b", 1]] "3.2.c_ac_ai" 3 2 [1, 2, -2, -8, -4, 8, 8] 0 [["1.2.c", 1], ["2.2.a_ae", 1]] "3.2.c_ab_ag" 3 2 [1, 2, -1, -6, -2, 8, 8] 2 [["1.2.c", 1], ["2.2.a_ad", 1]] "3.2.c_a_ae" 3 2 [1, 2, 0, -4, 0, 8, 8] 0 [["1.2.c", 1], ["2.2.a_ac", 1]] "3.2.c_a_ad" 3 2 [1, 2, 0, -3, 0, 8, 8] 3 [["3.2.c_a_ad", 1]] "3.2.c_b_ac" 3 2 [1, 2, 1, -2, 2, 8, 8] 2 [["1.2.c", 1], ["2.2.a_ab", 1]] "3.2.c_b_ab" 3 2 [1, 2, 1, -1, 2, 8, 8] 3 [["3.2.c_b_ab", 1]] "3.2.c_b_a" 3 2 [1, 2, 1, 0, 2, 8, 8] 2 [["3.2.c_b_a", 1]] "3.2.c_c_a" 3 2 [1, 2, 2, 0, 4, 8, 8] 0 [["1.2.ac", 1], ["1.2.c", 2]] "3.2.c_c_b" 3 2 [1, 2, 2, 1, 4, 8, 8] 3 [["3.2.c_c_b", 1]] "3.2.c_c_c" 3 2 [1, 2, 2, 2, 4, 8, 8] 0 [["3.2.c_c_c", 1]] "3.2.c_c_d" 3 2 [1, 2, 2, 3, 4, 8, 8] 3 [["1.2.b", 1], ["2.2.b_ab", 1]] "3.2.c_d_c" 3 2 [1, 2, 3, 2, 6, 8, 8] 2 [["1.2.c", 1], ["2.2.a_b", 1]] "3.2.c_d_d" 3 2 [1, 2, 3, 3, 6, 8, 8] 3 [["3.2.c_d_d", 1]] "3.2.c_d_e" 3 2 [1, 2, 3, 4, 6, 8, 8] 2 [["1.2.b", 1], ["2.2.b_a", 1]] "3.2.c_d_f" 3 2 [1, 2, 3, 5, 6, 8, 8] 3 [["3.2.c_d_f", 1]] "3.2.c_d_g" 3 2 [1, 2, 3, 6, 6, 8, 8] 2 [["3.2.c_d_g", 1]] "3.2.c_e_e" 3 2 [1, 2, 4, 4, 8, 8, 8] 0 [["1.2.c", 1], ["2.2.a_c", 1]] "3.2.c_e_f" 3 2 [1, 2, 4, 5, 8, 8, 8] 3 [["1.2.b", 1], ["2.2.b_b", 1]] "3.2.c_e_g" 3 2 [1, 2, 4, 6, 8, 8, 8] 0 [["3.2.c_e_g", 1]] "3.2.c_e_h" 3 2 [1, 2, 4, 7, 8, 8, 8] 3 [["1.2.ab", 1], ["2.2.d_f", 1]] "3.2.c_e_i" 3 2 [1, 2, 4, 8, 8, 8, 8] 0 [["1.2.a", 1], ["2.2.c_c", 1]] "3.2.c_f_g" 3 2 [1, 2, 5, 6, 10, 8, 8] 2 [["1.2.ab", 1], ["1.2.b", 1], ["1.2.c", 1]] "3.2.c_f_h" 3 2 [1, 2, 5, 7, 10, 8, 8] 3 [["3.2.c_f_h", 1]] "3.2.c_f_i" 3 2 [1, 2, 5, 8, 10, 8, 8] 2 [["1.2.a", 1], ["2.2.c_d", 1]] "3.2.c_g_h" 3 2 [1, 2, 6, 7, 12, 8, 8] 3 [["1.2.b", 1], ["2.2.b_d", 1]] "3.2.c_g_i" 3 2 [1, 2, 6, 8, 12, 8, 8] 0 [["1.2.a", 2], ["1.2.c", 1]] "3.2.c_h_i" 3 2 [1, 2, 7, 8, 14, 8, 8] 2 [["1.2.a", 1], ["1.2.b", 2]] "3.2.d_c_ab" 3 2 [1, 3, 2, -1, 4, 12, 8] 3 [["3.2.d_c_ab", 1]] "3.2.d_d_c" 3 2 [1, 3, 3, 2, 6, 12, 8] 2 [["1.2.c", 1], ["2.2.b_ab", 1]] "3.2.d_e_e" 3 2 [1, 3, 4, 4, 8, 12, 8] 1 [["1.2.c", 1], ["2.2.b_a", 1]] "3.2.d_f_g" 3 2 [1, 3, 5, 6, 10, 12, 8] 2 [["1.2.c", 1], ["2.2.b_b", 1]] "3.2.d_f_h" 3 2 [1, 3, 5, 7, 10, 12, 8] 3 [["3.2.d_f_h", 1]] "3.2.d_g_i" 3 2 [1, 3, 6, 8, 12, 12, 8] 1 [["1.2.ab", 1], ["1.2.c", 2]] "3.2.d_g_j" 3 2 [1, 3, 6, 9, 12, 12, 8] 3 [["3.2.d_g_j", 1]] "3.2.d_g_k" 3 2 [1, 3, 6, 10, 12, 12, 8] 1 [["1.2.b", 1], ["2.2.c_c", 1]] "3.2.d_h_k" 3 2 [1, 3, 7, 10, 14, 12, 8] 2 [["1.2.c", 1], ["2.2.b_d", 1]] "3.2.d_h_l" 3 2 [1, 3, 7, 11, 14, 12, 8] 3 [["1.2.b", 1], ["2.2.c_d", 1]] "3.2.d_h_m" 3 2 [1, 3, 7, 12, 14, 12, 8] 2 [["1.2.a", 1], ["2.2.d_f", 1]] "3.2.d_i_m" 3 2 [1, 3, 8, 12, 16, 12, 8] 1 [["1.2.a", 1], ["1.2.b", 1], ["1.2.c", 1]] "3.2.d_j_n" 3 2 [1, 3, 9, 13, 18, 12, 8] 3 [["1.2.b", 3]] "3.2.e_i_m" 3 2 [1, 4, 8, 12, 16, 16, 8] 0 [["1.2.c", 1], ["2.2.c_c", 1]] "3.2.e_j_o" 3 2 [1, 4, 9, 14, 18, 16, 8] 2 [["1.2.c", 1], ["2.2.c_d", 1]] "3.2.e_j_p" 3 2 [1, 4, 9, 15, 18, 16, 8] 3 [["3.2.e_j_p", 1]] "3.2.e_k_q" 3 2 [1, 4, 10, 16, 20, 16, 8] 0 [["1.2.a", 1], ["1.2.c", 2]] "3.2.e_k_r" 3 2 [1, 4, 10, 17, 20, 16, 8] 3 [["1.2.b", 1], ["2.2.d_f", 1]] "3.2.e_l_s" 3 2 [1, 4, 11, 18, 22, 16, 8] 2 [["1.2.b", 2], ["1.2.c", 1]] "3.2.f_n_w" 3 2 [1, 5, 13, 22, 26, 20, 8] 2 [["1.2.c", 1], ["2.2.d_f", 1]] "3.2.f_o_y" 3 2 [1, 5, 14, 24, 28, 20, 8] 1 [["1.2.b", 1], ["1.2.c", 2]] "3.2.g_s_bg" 3 2 [1, 6, 18, 32, 36, 24, 8] 0 [["1.2.c", 3]] # Label -- # The LMFDB uses a systematic system to label # isogeny classes, # endomorphism ring, # weak equivalence classes, # Deligne module, and # polarizations defined over finite fields. # The **label** format for an isogeny class defined over a finite field is **g.q.isog**, where # - **$g$** is the dimension of the abelian varieties contained in the isogeny class, # - **$q$** is the cardinality of the field over which the abelian varieties and the isogenies are defined, and # - **isog** specifies the isogeny class. # The label **isog** is obtained in the following manner: If the Weil $q$-polynomial of the isogeny class is # $$1 + a_1 x + a_2 x^2 + \cdots +a_gx^g+ qa_{g-1}x^{g+1} \cdots + q^{g-1}a_1 x^{2g-1} + q^g x^{2g},$$ # the label contains the integer coefficients $a_1, \ldots a_g$, encoded in base 26 with the symbols a, b, c... z, where a = 0, and separated by underscores. Negative numbers are distinguished from positive numbers by a leading a. For example, ae_j_ap denotes the polynomial # $$1 -4x + 9x^2 - 15x^3 +9qx^4 - 4q^2 x^5 + q^3 x^6,$$ # where $q$ is the cardinality of the field, because e = 4, j = 9 and p = 15. # Each endomorphism ring of an ordinary abelian variety has a label of the form **g.q.isog.N.i**, where # - **$N :=[\mathcal{O}_{\mathbb{Q}[F]}:R]$** is the index of the endomorphism ring $R$ in the maximal order $\mathcal{O}_{\mathbb{Q}[F]}$ of the field generated by the Frobenius endomorphism $\mathbb{Q}[F]$, # - **$i$** is an index that uniquely determines the endomorphism ring among all the overorders of $\Z[F,V]$ with index $N$ in $\mathcal{O}_{\mathbb{Q}[F]}$. # Each weak equivalence class of an ordinary isogeny class has a label of the form **g.q.isog.N.i.w**, where # - **$w$** is an index that uniquely determines the weak equivalence class class among all the with the same endomorphism ring. # Each Deligne module of an ordinary isogeny class has a label of the form **g.q.isog.N.i.w.j**, where # - **$j$** is an index that uniquely determines the isomorphism class of the Deligne module representing the unpolarized abelian variety within the same weak equivalence class. # Each polarizations of an ordinary isogeny class has a label of the form **g.q.isog.N.i.w.j.d.k**, where # - **$d$** is the degree of the polarization, # - **$k$** is an index that determines the polarization among all the ones with the same degree. #Dimension (g) -- # The **dimension** of an algebraic variety $V$ is the maximal length $d$ of a chain # $$ # V_0 \subset V_1 \subset \cdots \subset V_d # $$ # of distinct irreducible subvarieties of $V$. #Base field (q) -- # The **base field**, of an algebraic variety is the field over which it is defined; it necessarily contains the coefficients of a set of defining equations for the variety, but it is not necessarily a minimal field of definition. #L-polynomial (polynomial) -- # Let $A$ be an abelian variety of dimension $g$ defined over $\F_q$. Let $F_q$ be the inverse of the field automorphism $x \mapsto x^q$ in $\Gal(\overline{\F}_q/\F_q)$, which acts on $\ell$-adic \'etale cohomology. The **L-polynomial** of $A$ is # $$L_A(t) = \det(1-t F_q|H^1(A_{\overline{\F}_q}, \Q_\ell)).$$ # This is a polynomial of degree $2g$ with integer coefficients that are independent of $\ell$. Its constant term is $1$. # The L-polynomial $L_A(t)$ is the reverse of the characteristic polynomial $P_A(t)$, which is a Weil $q$-polynomial. Thus the complex roots of $L_A(t)$ have absolute value $q^{-1/2}$. #$p$-rank (p_rank) -- # Let $A$ be a $g$-dimensional abelian variety over $\F_q$ where $q=p^r$. # The **$p$-rank** of $A$ is the dimension of the geometric $p$-torsion as an $\F_p$-vector space: $$p\operatorname{-rank}(A) = \dim_{\F_p}( A(\overline{\F}_p)[p] ).$$ The $p$-rank is at most $g$, with equality if and only if $A$ is ordinary. The difference between $g$ and the $p$-rank is the **$p$-corank** of $A$. #Isogeny factors (decompositionraw) -- # Any abelian variety $A$ is isogenous to a product of simple abelian varieties $B_i$, called the **isogeny factors** of $A$: # $$A \sim B_1 \times \cdots \times B_n.$$ # We say that $A$ decomposes up to isogeny into the product of the $B_i$. # Note that two elements of this product might be isogenous; in other words, elements of the decomposition may appear with multiplicity.