# Properties

 Label 6.2.ak_ca_agz_ry_abju_ceq Base Field $\F_{2}$ Dimension $6$ Ordinary No $p$-rank $3$ Principally polarizable Yes Contains a Jacobian No

## Invariants

 Base field: $\F_{2}$ Dimension: $6$ L-polynomial: $( 1 - x + 2 x^{2} )( 1 - 2 x + 2 x^{2} )^{3}( 1 - 3 x + 5 x^{2} - 6 x^{3} + 4 x^{4} )$ Frobenius angles: $\pm0.123548644961$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.384973271919$, $\pm0.456881978294$ Angle rank: $2$ (numerical)

This isogeny class is not simple.

## Newton polygon

 $p$-rank: $3$ Slopes: $[0, 0, 0, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1, 1, 1]$

## Point counts

This isogeny class is principally polarizable, but does not contain a Jacobian.

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 2 19000 2337608 42750000 1457127782 88829104000 4518059688074 229434889500000 15066443835547832 1161057630794875000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -7 9 26 33 43 78 133 209 422 1029

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
 The isogeny class factors as 1.2.ac 3 $\times$ 1.2.ab $\times$ 2.2.ad_f and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: 1.2.ac 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-1})$$$)$ 1.2.ab : $$\Q(\sqrt{-7})$$. 2.2.ad_f : $$\Q(\sqrt{-3}, \sqrt{5})$$.
Endomorphism algebra over $\overline{\F}_{2}$
 The base change of $A$ to $\F_{2^{12}}$ is 1.4096.h 2 $\times$ 1.4096.bv $\times$ 1.4096.ey 3 . The endomorphism algebra for each factor is: 1.4096.h 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-15})$$$)$ 1.4096.bv : $$\Q(\sqrt{-7})$$. 1.4096.ey 3 : $\mathrm{M}_{3}(B)$, where $B$ is the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$.
All geometric endomorphisms are defined over $\F_{2^{12}}$.
Remainder of endomorphism lattice by field
• Endomorphism algebra over $\F_{2^{2}}$  The base change of $A$ to $\F_{2^{2}}$ is 1.4.a 3 $\times$ 1.4.d $\times$ 2.4.b_ad. The endomorphism algebra for each factor is: 1.4.a 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-1})$$$)$ 1.4.d : $$\Q(\sqrt{-7})$$. 2.4.b_ad : $$\Q(\sqrt{-3}, \sqrt{5})$$.
• Endomorphism algebra over $\F_{2^{3}}$  The base change of $A$ to $\F_{2^{3}}$ is 1.8.e 3 $\times$ 1.8.f $\times$ 2.8.a_l. The endomorphism algebra for each factor is: 1.8.e 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-1})$$$)$ 1.8.f : $$\Q(\sqrt{-7})$$. 2.8.a_l : $$\Q(\sqrt{-3}, \sqrt{5})$$.
• Endomorphism algebra over $\F_{2^{4}}$  The base change of $A$ to $\F_{2^{4}}$ is 1.16.ab $\times$ 1.16.i 3 $\times$ 2.16.ah_bh. The endomorphism algebra for each factor is: 1.16.ab : $$\Q(\sqrt{-7})$$. 1.16.i 3 : $\mathrm{M}_{3}(B)$, where $B$ is the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$. 2.16.ah_bh : $$\Q(\sqrt{-3}, \sqrt{5})$$.
• Endomorphism algebra over $\F_{2^{6}}$  The base change of $A$ to $\F_{2^{6}}$ is 1.64.aj $\times$ 1.64.a 3 $\times$ 1.64.l 2 . The endomorphism algebra for each factor is: 1.64.aj : $$\Q(\sqrt{-7})$$. 1.64.a 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-1})$$$)$ 1.64.l 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-15})$$$)$

## Base change

This is a primitive isogeny class.

## Twists

Below are some of the twists of this isogeny class.
 Twist Extension Degree Common base change 6.2.ai_bi_adv_io_apq_ya $2$ (not in LMFDB) 6.2.ag_u_abt_da_aek_ge $2$ (not in LMFDB) 6.2.ae_k_at_bi_acc_dc $2$ (not in LMFDB) 6.2.ae_k_an_k_g_aq $2$ (not in LMFDB) 6.2.ac_e_af_k_as_bg $2$ (not in LMFDB) 6.2.ac_e_ad_g_c_a $2$ (not in LMFDB) 6.2.a_c_ad_g_ag_q $2$ (not in LMFDB) 6.2.a_c_d_g_g_q $2$ (not in LMFDB) 6.2.c_e_d_g_ac_a $2$ (not in LMFDB) 6.2.c_e_f_k_s_bg $2$ (not in LMFDB) 6.2.e_k_n_k_ag_aq $2$ (not in LMFDB) 6.2.e_k_t_bi_cc_dc $2$ (not in LMFDB) 6.2.g_u_bt_da_ek_ge $2$ (not in LMFDB) 6.2.i_bi_dv_io_pq_ya $2$ (not in LMFDB) 6.2.k_ca_gz_ry_bju_ceq $2$ (not in LMFDB) 6.2.ah_z_acd_de_adm_ea $3$ (not in LMFDB) 6.2.ae_k_an_e_y_aca $3$ (not in LMFDB) 6.2.ab_b_f_ac_a_u $3$ (not in LMFDB) 6.2.c_e_l_q_y_bs $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 6.2.ai_bi_adv_io_apq_ya $2$ (not in LMFDB) 6.2.ag_u_abt_da_aek_ge $2$ (not in LMFDB) 6.2.ae_k_at_bi_acc_dc $2$ (not in LMFDB) 6.2.ae_k_an_k_g_aq $2$ (not in LMFDB) 6.2.ac_e_af_k_as_bg $2$ (not in LMFDB) 6.2.ac_e_ad_g_c_a $2$ (not in LMFDB) 6.2.a_c_ad_g_ag_q $2$ (not in LMFDB) 6.2.a_c_d_g_g_q $2$ (not in LMFDB) 6.2.c_e_d_g_ac_a $2$ (not in LMFDB) 6.2.c_e_f_k_s_bg $2$ (not in LMFDB) 6.2.e_k_n_k_ag_aq $2$ (not in LMFDB) 6.2.e_k_t_bi_cc_dc $2$ (not in LMFDB) 6.2.g_u_bt_da_ek_ge $2$ (not in LMFDB) 6.2.i_bi_dv_io_pq_ya $2$ (not in LMFDB) 6.2.k_ca_gz_ry_bju_ceq $2$ (not in LMFDB) 6.2.ah_z_acd_de_adm_ea $3$ (not in LMFDB) 6.2.ae_k_an_e_y_aca $3$ (not in LMFDB) 6.2.ab_b_f_ac_a_u $3$ (not in LMFDB) 6.2.c_e_l_q_y_bs $3$ (not in LMFDB) 6.2.c_e_d_g_ac_a $4$ (not in LMFDB) 6.2.ai_bi_adx_iy_aqq_zs $6$ (not in LMFDB) 6.2.ag_u_abz_ee_ahk_lg $6$ (not in LMFDB) 6.2.af_n_ax_bi_abw_cq $6$ (not in LMFDB) 6.2.af_n_av_be_abu_cu $6$ (not in LMFDB) 6.2.ae_k_av_bk_ace_dg $6$ (not in LMFDB) 6.2.ad_f_aj_s_ay_bc $6$ (not in LMFDB) 6.2.ad_f_ad_g_as_bo $6$ (not in LMFDB) 6.2.ac_e_al_q_ay_bs $6$ (not in LMFDB) 6.2.ac_e_af_e_a_ae $6$ (not in LMFDB) 6.2.ac_e_ad_a_i_am $6$ (not in LMFDB) 6.2.ab_b_ad_g_ai_m $6$ (not in LMFDB) 6.2.ab_b_ab_k_ag_i $6$ (not in LMFDB) 6.2.a_c_ad_a_a_e $6$ (not in LMFDB) 6.2.a_c_d_a_a_e $6$ (not in LMFDB) 6.2.b_b_af_ac_a_u $6$ (not in LMFDB) 6.2.b_b_b_k_g_i $6$ (not in LMFDB) 6.2.b_b_d_g_i_m $6$ (not in LMFDB) 6.2.c_e_d_a_ai_am $6$ (not in LMFDB) 6.2.c_e_d_g_ac_a $6$ (not in LMFDB) 6.2.c_e_f_e_a_ae $6$ (not in LMFDB) 6.2.d_f_d_g_s_bo $6$ (not in LMFDB) 6.2.d_f_j_s_y_bc $6$ (not in LMFDB) 6.2.e_k_n_e_ay_aca $6$ (not in LMFDB) 6.2.e_k_v_bk_ce_dg $6$ (not in LMFDB) 6.2.f_n_v_be_bu_cu $6$ (not in LMFDB) 6.2.f_n_x_bi_bw_cq $6$ (not in LMFDB) 6.2.g_u_bz_ee_hk_lg $6$ (not in LMFDB) 6.2.h_z_cd_de_dm_ea $6$ (not in LMFDB) 6.2.i_bi_dx_iy_qq_zs $6$ (not in LMFDB) 6.2.ai_bk_aej_km_auc_bfk $8$ (not in LMFDB) 6.2.ag_q_av_ac_co_afg $8$ (not in LMFDB) 6.2.ag_w_ach_ey_aiw_no $8$ (not in LMFDB) 6.2.ag_y_acr_gc_ali_ro $8$ (not in LMFDB) 6.2.ae_g_ad_ag_w_abo $8$ (not in LMFDB) 6.2.ae_i_aj_ae_bi_acm $8$ (not in LMFDB) 6.2.ae_m_az_bs_aco_ds $8$ (not in LMFDB) 6.2.ae_o_abj_cw_afa_hs $8$ (not in LMFDB) 6.2.ae_q_abp_do_agk_jw $8$ (not in LMFDB) 6.2.ac_a_d_ag_c_i $8$ (not in LMFDB) 6.2.ac_c_ad_ae_o_aq $8$ (not in LMFDB) 6.2.ac_g_al_u_abe_bw $8$ (not in LMFDB) 6.2.ac_g_af_i_g_a $8$ (not in LMFDB) 6.2.ac_i_an_ba_abm_ce $8$ (not in LMFDB) 6.2.ac_k_at_bs_acw_ei $8$ (not in LMFDB) 6.2.a_ac_ad_ac_g_i $8$ (not in LMFDB) 6.2.a_ac_d_ac_ag_i $8$ (not in LMFDB) 6.2.a_e_ab_i_ak_q $8$ (not in LMFDB) 6.2.a_e_b_i_k_q $8$ (not in LMFDB) 6.2.a_g_ad_o_as_y $8$ (not in LMFDB) 6.2.a_g_d_o_s_y $8$ (not in LMFDB) 6.2.c_a_ad_ag_ac_i $8$ (not in LMFDB) 6.2.c_c_d_ae_ao_aq $8$ (not in LMFDB) 6.2.c_g_f_i_ag_a $8$ (not in LMFDB) 6.2.c_g_l_u_be_bw $8$ (not in LMFDB) 6.2.c_i_n_ba_bm_ce $8$ (not in LMFDB) 6.2.c_k_t_bs_cw_ei $8$ (not in LMFDB) 6.2.e_g_d_ag_aw_abo $8$ (not in LMFDB) 6.2.e_i_j_ae_abi_acm $8$ (not in LMFDB) 6.2.e_m_z_bs_co_ds $8$ (not in LMFDB) 6.2.e_o_bj_cw_fa_hs $8$ (not in LMFDB) 6.2.e_q_bp_do_gk_jw $8$ (not in LMFDB) 6.2.g_q_v_ac_aco_afg $8$ (not in LMFDB) 6.2.g_w_ch_ey_iw_no $8$ (not in LMFDB) 6.2.g_y_cr_gc_li_ro $8$ (not in LMFDB) 6.2.i_bk_ej_km_uc_bfk $8$ (not in LMFDB) 6.2.ah_bb_acr_fe_aig_ma $12$ (not in LMFDB) 6.2.af_p_abh_ck_aea_ga $12$ (not in LMFDB) 6.2.af_p_abf_cg_adu_fw $12$ (not in LMFDB) 6.2.ad_h_ap_be_abw_cq $12$ (not in LMFDB) 6.2.ad_h_aj_s_abe_ce $12$ (not in LMFDB) 6.2.ab_d_af_k_aq_u $12$ (not in LMFDB) 6.2.ab_d_ad_o_ak_y $12$ (not in LMFDB) 6.2.ab_d_d_c_i_m $12$ (not in LMFDB) 6.2.b_d_ad_c_ai_m $12$ (not in LMFDB) 6.2.b_d_d_o_k_y $12$ (not in LMFDB) 6.2.b_d_f_k_q_u $12$ (not in LMFDB) 6.2.d_h_j_s_be_ce $12$ (not in LMFDB) 6.2.d_h_p_be_bw_cq $12$ (not in LMFDB) 6.2.f_p_bf_cg_du_fw $12$ (not in LMFDB) 6.2.f_p_bh_ck_ea_ga $12$ (not in LMFDB) 6.2.h_bb_cr_fe_ig_ma $12$ (not in LMFDB) 6.2.ag_s_abh_bm_ay_m $24$ (not in LMFDB) 6.2.ag_w_acj_fe_ajk_om $24$ (not in LMFDB) 6.2.ag_w_acf_eo_ahw_lw $24$ (not in LMFDB) 6.2.af_p_abd_bs_acc_cu $24$ (not in LMFDB) 6.2.af_r_abn_cy_aes_hc $24$ (not in LMFDB) 6.2.ae_i_al_o_aq_u $24$ (not in LMFDB) 6.2.ae_k_ar_u_aq_q $24$ (not in LMFDB) 6.2.ae_m_abf_ck_aee_gm $24$ (not in LMFDB) 6.2.ae_m_abb_cc_ado_fk $24$ (not in LMFDB) 6.2.ae_o_abh_cq_aem_gu $24$ (not in LMFDB) 6.2.ad_b_j_ao_ag_bg $24$ (not in LMFDB) 6.2.ad_d_d_ak_g_a $24$ (not in LMFDB) 6.2.ad_d_d_ae_am_bk $24$ (not in LMFDB) 6.2.ad_f_ad_e_am_bc $24$ (not in LMFDB) 6.2.ad_h_an_u_abc_bo $24$ (not in LMFDB) 6.2.ad_h_al_u_aba_bo $24$ (not in LMFDB) 6.2.ad_h_aj_q_ay_bs $24$ (not in LMFDB) 6.2.ad_j_at_bk_aci_dk $24$ (not in LMFDB) 6.2.ad_j_ar_bk_acc_dk $24$ (not in LMFDB) 6.2.ad_j_ap_ba_abe_bw $24$ (not in LMFDB) 6.2.ad_j_ap_bg_abw_dg $24$ (not in LMFDB) 6.2.ad_l_av_bu_aco_ei $24$ (not in LMFDB) 6.2.ac_c_ab_c_ai_u $24$ (not in LMFDB) 6.2.ac_e_ah_i_ai_q $24$ (not in LMFDB) 6.2.ac_g_aj_s_abc_bs $24$ (not in LMFDB) 6.2.ac_g_af_c_m_ay $24$ (not in LMFDB) 6.2.ac_i_ap_bg_aca_dc $24$ (not in LMFDB) 6.2.ab_ad_d_g_ac_aq $24$ (not in LMFDB) 6.2.ab_ab_b_c_c_aq $24$ (not in LMFDB) 6.2.ab_ab_b_i_ae_ae $24$ (not in LMFDB) 6.2.ab_ab_d_ae_ac_i $24$ (not in LMFDB) 6.2.ab_b_ab_i_ae_e $24$ (not in LMFDB) 6.2.ab_b_b_ae_c_ai $24$ (not in LMFDB) 6.2.ab_b_b_c_ae_q $24$ (not in LMFDB) 6.2.ab_d_ah_i_am_y $24$ (not in LMFDB) 6.2.ab_d_ad_m_ai_u $24$ (not in LMFDB) 6.2.ab_d_ab_g_ae_q $24$ (not in LMFDB) 6.2.ab_d_ab_i_ag_y $24$ (not in LMFDB) 6.2.ab_f_aj_q_abc_bo $24$ (not in LMFDB) 6.2.ab_f_af_o_ak_bg $24$ (not in LMFDB) 6.2.ab_f_af_u_aq_bs $24$ (not in LMFDB) 6.2.ab_f_ad_o_ai_bg $24$ (not in LMFDB) 6.2.ab_f_ad_q_ak_bo $24$ (not in LMFDB) 6.2.ab_h_ah_ba_aw_cm $24$ (not in LMFDB) 6.2.ab_h_af_u_ak_bo $24$ (not in LMFDB) 6.2.ab_h_af_ba_aq_cm $24$ (not in LMFDB) 6.2.ab_j_ah_bk_aw_dk $24$ (not in LMFDB) 6.2.a_a_ad_c_a_m $24$ (not in LMFDB) 6.2.a_a_d_c_a_m $24$ (not in LMFDB) 6.2.a_e_ad_k_am_u $24$ (not in LMFDB) 6.2.a_e_ab_c_ae_ai $24$ (not in LMFDB) 6.2.a_e_b_c_e_ai $24$ (not in LMFDB) 6.2.a_e_d_k_m_u $24$ (not in LMFDB) 6.2.b_ad_ad_g_c_aq $24$ (not in LMFDB) 6.2.b_ab_ad_ae_c_i $24$ (not in LMFDB) 6.2.b_ab_ab_c_ac_aq $24$ (not in LMFDB) 6.2.b_ab_ab_i_e_ae $24$ (not in LMFDB) 6.2.b_b_ab_ae_ac_ai $24$ (not in LMFDB) 6.2.b_b_ab_c_e_q $24$ (not in LMFDB) 6.2.b_b_b_i_e_e $24$ (not in LMFDB) 6.2.b_d_b_g_e_q $24$ (not in LMFDB) 6.2.b_d_b_i_g_y $24$ (not in LMFDB) 6.2.b_d_d_m_i_u $24$ (not in LMFDB) 6.2.b_d_h_i_m_y $24$ (not in LMFDB) 6.2.b_f_d_o_i_bg $24$ (not in LMFDB) 6.2.b_f_d_q_k_bo $24$ (not in LMFDB) 6.2.b_f_f_o_k_bg $24$ (not in LMFDB) 6.2.b_f_f_u_q_bs $24$ (not in LMFDB) 6.2.b_f_j_q_bc_bo $24$ (not in LMFDB) 6.2.b_h_f_u_k_bo $24$ (not in LMFDB) 6.2.b_h_f_ba_q_cm $24$ (not in LMFDB) 6.2.b_h_h_ba_w_cm $24$ (not in LMFDB) 6.2.b_j_h_bk_w_dk $24$ (not in LMFDB) 6.2.c_c_b_c_i_u $24$ (not in LMFDB) 6.2.c_e_h_i_i_q $24$ (not in LMFDB) 6.2.c_g_f_c_am_ay $24$ (not in LMFDB) 6.2.c_g_j_s_bc_bs $24$ (not in LMFDB) 6.2.c_i_p_bg_ca_dc $24$ (not in LMFDB) 6.2.d_b_aj_ao_g_bg $24$ (not in LMFDB) 6.2.d_d_ad_ak_ag_a $24$ (not in LMFDB) 6.2.d_d_ad_ae_m_bk $24$ (not in LMFDB) 6.2.d_f_d_e_m_bc $24$ (not in LMFDB) 6.2.d_h_j_q_y_bs $24$ (not in LMFDB) 6.2.d_h_l_u_ba_bo $24$ (not in LMFDB) 6.2.d_h_n_u_bc_bo $24$ (not in LMFDB) 6.2.d_j_p_ba_be_bw $24$ (not in LMFDB) 6.2.d_j_p_bg_bw_dg $24$ (not in LMFDB) 6.2.d_j_r_bk_cc_dk $24$ (not in LMFDB) 6.2.d_j_t_bk_ci_dk $24$ (not in LMFDB) 6.2.d_l_v_bu_co_ei $24$ (not in LMFDB) 6.2.e_i_l_o_q_u $24$ (not in LMFDB) 6.2.e_k_r_u_q_q $24$ (not in LMFDB) 6.2.e_m_bb_cc_do_fk $24$ (not in LMFDB) 6.2.e_m_bf_ck_ee_gm $24$ (not in LMFDB) 6.2.e_o_bh_cq_em_gu $24$ (not in LMFDB) 6.2.f_p_bd_bs_cc_cu $24$ (not in LMFDB) 6.2.f_r_bn_cy_es_hc $24$ (not in LMFDB) 6.2.g_s_bh_bm_y_m $24$ (not in LMFDB) 6.2.g_w_cf_eo_hw_lw $24$ (not in LMFDB) 6.2.g_w_cj_fe_jk_om $24$ (not in LMFDB)