# Properties

 Label 6.2.ai_be_acs_en_agf_ik Base Field $\F_{2}$ Dimension $6$ Ordinary No $p$-rank $5$ Principally polarizable Yes Contains a Jacobian No

## Invariants

 Base field: $\F_{2}$ Dimension: $6$ L-polynomial: $( 1 - 2 x + 2 x^{2} )( 1 - 3 x + 5 x^{2} - 6 x^{3} + 4 x^{4} )( 1 - 3 x + 2 x^{2} + x^{3} + 4 x^{4} - 12 x^{5} + 8 x^{6} )$ Frobenius angles: $\pm0.0992589862044$, $\pm0.123548644961$, $\pm0.186455299510$, $\pm0.250000000000$, $\pm0.456881978294$, $\pm0.757883870938$ Angle rank: $2$ (numerical)

This isogeny class is not simple.

## Newton polygon

 $p$-rank: $5$ Slopes: $[0, 0, 0, 0, 0, 1/2, 1/2, 1, 1, 1, 1, 1]$

## 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 $A(\F_{q^r})$ 1 2755 297388 34836975 1373558261 140920277680 7134635096744 258649803334575 18833275237588084 1213029167817625525

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -5 1 7 29 40 109 191 237 538 1076

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
 The isogeny class factors as 1.2.ac $\times$ 2.2.ad_f $\times$ 3.2.ad_c_b and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
Endomorphism algebra over $\overline{\F}_{2}$
 The base change of $A$ to $\F_{2^{84}}$ is 1.19342813113834066795298816.auojdvfkpl 3 $\times$ 1.19342813113834066795298816.aptgzwqopl 2 $\times$ 1.19342813113834066795298816.bqdecdesiy. The endomorphism algebra for each factor is: 1.19342813113834066795298816.auojdvfkpl 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-7})$$$)$ 1.19342813113834066795298816.aptgzwqopl 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-15})$$$)$ 1.19342813113834066795298816.bqdecdesiy : the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$.
All geometric endomorphisms are defined over $\F_{2^{84}}$.
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 $\times$ 2.4.b_ad $\times$ 3.4.af_s_abp. The endomorphism algebra for each factor is:
• Endomorphism algebra over $\F_{2^{3}}$  The base change of $A$ to $\F_{2^{3}}$ is 1.8.e $\times$ 2.8.a_l $\times$ 3.8.ag_bd_adf. The endomorphism algebra for each factor is:
• Endomorphism algebra over $\F_{2^{4}}$  The base change of $A$ to $\F_{2^{4}}$ is 1.16.i $\times$ 2.16.ah_bh $\times$ 3.16.l_cg_jf. The endomorphism algebra for each factor is: 1.16.i : the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$. 2.16.ah_bh : $$\Q(\sqrt{-3}, \sqrt{5})$$. 3.16.l_cg_jf : $$\Q(\zeta_{7})$$.
• Endomorphism algebra over $\F_{2^{6}}$  The base change of $A$ to $\F_{2^{6}}$ is 1.64.a $\times$ 1.64.l 2 $\times$ 3.64.w_lx_ekt. The endomorphism algebra for each factor is: 1.64.a : $$\Q(\sqrt{-1})$$. 1.64.l 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-15})$$$)$ 3.64.w_lx_ekt : $$\Q(\zeta_{7})$$.
• Endomorphism algebra over $\F_{2^{7}}$  The base change of $A$ to $\F_{2^{7}}$ is 1.128.aq $\times$ 1.128.n 3 $\times$ 2.128.bn_yl. The endomorphism algebra for each factor is: 1.128.aq : $$\Q(\sqrt{-1})$$. 1.128.n 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-7})$$$)$ 2.128.bn_yl : $$\Q(\sqrt{-3}, \sqrt{5})$$.
• Endomorphism algebra over $\F_{2^{12}}$  The base change of $A$ to $\F_{2^{12}}$ is 1.4096.h 2 $\times$ 1.4096.ey $\times$ 3.4096.fe_fpl_agrpx. The endomorphism algebra for each factor is: 1.4096.h 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-15})$$$)$ 1.4096.ey : the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$. 3.4096.fe_fpl_agrpx : $$\Q(\zeta_{7})$$.
• Endomorphism algebra over $\F_{2^{14}}$  The base change of $A$ to $\F_{2^{14}}$ is 1.16384.a $\times$ 1.16384.dj 3 $\times$ 2.16384.ajr_cqyz. The endomorphism algebra for each factor is: 1.16384.a : $$\Q(\sqrt{-1})$$. 1.16384.dj 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-7})$$$)$ 2.16384.ajr_cqyz : $$\Q(\sqrt{-3}, \sqrt{5})$$.
• Endomorphism algebra over $\F_{2^{21}}$  The base change of $A$ to $\F_{2^{21}}$ is 1.2097152.aedn 3 $\times$ 1.2097152.dau $\times$ 2.2097152.a_hpued. The endomorphism algebra for each factor is: 1.2097152.aedn 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-7})$$$)$ 1.2097152.dau : $$\Q(\sqrt{-1})$$. 2.2097152.a_hpued : $$\Q(\sqrt{-3}, \sqrt{5})$$.
• Endomorphism algebra over $\F_{2^{28}}$  The base change of $A$ to $\F_{2^{28}}$ is 1.268435456.blhf 3 $\times$ 1.268435456.bwmi $\times$ 2.268435456.bssv_ccitvwj. The endomorphism algebra for each factor is: 1.268435456.blhf 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-7})$$$)$ 1.268435456.bwmi : the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$. 2.268435456.bssv_ccitvwj : $$\Q(\sqrt{-3}, \sqrt{5})$$.
• Endomorphism algebra over $\F_{2^{42}}$  The base change of $A$ to $\F_{2^{42}}$ is 1.4398046511104.ahxvrd 3 $\times$ 1.4398046511104.a $\times$ 1.4398046511104.hpued 2 . The endomorphism algebra for each factor is: 1.4398046511104.ahxvrd 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-7})$$$)$ 1.4398046511104.a : $$\Q(\sqrt{-1})$$. 1.4398046511104.hpued 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.ae_g_ag_n_abd_bu $2$ (not in LMFDB) 6.2.ac_a_c_d_af_c $2$ (not in LMFDB) 6.2.ac_a_g_af_ah_w $2$ (not in LMFDB) 6.2.c_a_ag_af_h_w $2$ (not in LMFDB) 6.2.c_a_ac_d_f_c $2$ (not in LMFDB) 6.2.e_g_g_n_bd_bu $2$ (not in LMFDB) 6.2.i_be_cs_en_gf_ik $2$ (not in LMFDB) 6.2.af_j_ae_a_abd_cw $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 6.2.ae_g_ag_n_abd_bu $2$ (not in LMFDB) 6.2.ac_a_c_d_af_c $2$ (not in LMFDB) 6.2.ac_a_g_af_ah_w $2$ (not in LMFDB) 6.2.c_a_ag_af_h_w $2$ (not in LMFDB) 6.2.c_a_ac_d_f_c $2$ (not in LMFDB) 6.2.e_g_g_n_bd_bu $2$ (not in LMFDB) 6.2.i_be_cs_en_gf_ik $2$ (not in LMFDB) 6.2.af_j_ae_a_abd_cw $3$ (not in LMFDB) 6.2.ab_ad_a_q_af_aba $6$ (not in LMFDB) 6.2.b_ad_a_q_f_aba $6$ (not in LMFDB) 6.2.f_j_e_a_bd_cw $6$ (not in LMFDB) 6.2.ai_bl_aep_lg_avv_big $7$ (not in LMFDB) 6.2.ab_c_a_ac_h_ag $7$ (not in LMFDB) 6.2.ag_s_abm_cn_adr_fc $8$ (not in LMFDB) 6.2.a_a_ac_ab_b_m $8$ (not in LMFDB) 6.2.a_a_c_ab_ab_m $8$ (not in LMFDB) 6.2.g_s_bm_cn_dr_fc $8$ (not in LMFDB) 6.2.af_l_ao_u_abv_di $12$ (not in LMFDB) 6.2.ab_ab_ac_m_ah_ag $12$ (not in LMFDB) 6.2.b_ab_c_m_h_ag $12$ (not in LMFDB) 6.2.f_l_o_u_bv_di $12$ (not in LMFDB) 6.2.aj_bq_afe_mm_axz_blm $14$ (not in LMFDB) 6.2.ag_x_acl_fi_ajp_os $14$ (not in LMFDB) 6.2.af_o_abe_ca_adb_ek $14$ (not in LMFDB) 6.2.ae_n_abf_cm_aeh_go $14$ (not in LMFDB) 6.2.ae_n_abb_bw_act_dy $14$ (not in LMFDB) 6.2.ad_g_ai_g_b_ak $14$ (not in LMFDB) 6.2.ac_h_ap_be_abx_da $14$ (not in LMFDB) 6.2.ac_h_al_w_abh_by $14$ (not in LMFDB) 6.2.ac_h_af_k_j_c $14$ (not in LMFDB) 6.2.a_f_ad_m_ap_w $14$ (not in LMFDB) 6.2.a_f_d_m_p_w $14$ (not in LMFDB) 6.2.b_c_a_ac_ah_ag $14$ (not in LMFDB) 6.2.c_h_f_k_aj_c $14$ (not in LMFDB) 6.2.c_h_l_w_bh_by $14$ (not in LMFDB) 6.2.c_h_p_be_bx_da $14$ (not in LMFDB) 6.2.d_g_i_g_ab_ak $14$ (not in LMFDB) 6.2.e_n_bb_bw_ct_dy $14$ (not in LMFDB) 6.2.e_n_bf_cm_eh_go $14$ (not in LMFDB) 6.2.f_o_be_ca_db_ek $14$ (not in LMFDB) 6.2.g_x_cl_fi_jp_os $14$ (not in LMFDB) 6.2.i_bl_ep_lg_vv_big $14$ (not in LMFDB) 6.2.j_bq_fe_mm_xz_blm $14$ (not in LMFDB) 6.2.af_n_ar_b_bt_adq $21$ (not in LMFDB) 6.2.af_q_abg_bx_acf_cw $21$ (not in LMFDB) 6.2.ac_b_h_ai_ad_ba $21$ (not in LMFDB) 6.2.b_b_h_h_j_ba $21$ (not in LMFDB) 6.2.c_c_d_h_n_s $21$ (not in LMFDB) 6.2.ad_d_ac_i_ar_y $24$ (not in LMFDB) 6.2.ad_f_ai_q_abb_bo $24$ (not in LMFDB) 6.2.d_d_c_i_r_y $24$ (not in LMFDB) 6.2.d_f_i_q_bb_bo $24$ (not in LMFDB) 6.2.ag_r_abb_s_v_ack $28$ (not in LMFDB) 6.2.ae_h_ah_e_d_ak $28$ (not in LMFDB) 6.2.ac_b_b_ac_ad_o $28$ (not in LMFDB) 6.2.a_ab_ad_a_d_k $28$ (not in LMFDB) 6.2.a_ab_d_a_ad_k $28$ (not in LMFDB) 6.2.c_b_ab_ac_d_o $28$ (not in LMFDB) 6.2.e_h_h_e_ad_ak $28$ (not in LMFDB) 6.2.g_r_bb_s_av_ack $28$ (not in LMFDB) 6.2.ah_z_acj_el_ahb_kg $42$ (not in LMFDB) 6.2.ag_s_abj_bz_acn_di $42$ (not in LMFDB) 6.2.af_n_abb_bz_adh_ew $42$ (not in LMFDB) 6.2.ae_h_ah_k_abb_by $42$ (not in LMFDB) 6.2.ad_f_aj_p_abb_bu $42$ (not in LMFDB) 6.2.ad_f_ad_ad_p_aba $42$ (not in LMFDB) 6.2.ad_i_am_v_abb_bu $42$ (not in LMFDB) 6.2.ac_b_ad_m_an_g $42$ (not in LMFDB) 6.2.ac_c_ad_h_an_s $42$ (not in LMFDB) 6.2.ab_b_ah_h_aj_ba $42$ (not in LMFDB) 6.2.ab_b_ab_b_ad_c $42$ (not in LMFDB) 6.2.ab_b_d_ad_b_g $42$ (not in LMFDB) 6.2.ab_e_ae_n_aj_ba $42$ (not in LMFDB) 6.2.ab_e_a_j_af_be $42$ (not in LMFDB) 6.2.a_ab_ad_g_ad_ac $42$ (not in LMFDB) 6.2.a_ab_d_g_d_ac $42$ (not in LMFDB) 6.2.b_b_ad_ad_ab_g $42$ (not in LMFDB) 6.2.b_b_b_b_d_c $42$ (not in LMFDB) 6.2.b_e_a_j_f_be $42$ (not in LMFDB) 6.2.b_e_e_n_j_ba $42$ (not in LMFDB) 6.2.c_b_ah_ai_d_ba $42$ (not in LMFDB) 6.2.c_b_d_m_n_g $42$ (not in LMFDB) 6.2.d_f_d_ad_ap_aba $42$ (not in LMFDB) 6.2.d_f_j_p_bb_bu $42$ (not in LMFDB) 6.2.d_i_m_v_bb_bu $42$ (not in LMFDB) 6.2.e_h_h_k_bb_by $42$ (not in LMFDB) 6.2.f_n_r_b_abt_adq $42$ (not in LMFDB) 6.2.f_n_bb_bz_dh_ew $42$ (not in LMFDB) 6.2.f_q_bg_bx_cf_cw $42$ (not in LMFDB) 6.2.g_s_bj_bz_cn_di $42$ (not in LMFDB) 6.2.h_z_cj_el_hb_kg $42$ (not in LMFDB) 6.2.ah_bc_ade_hg_ann_uy $56$ (not in LMFDB) 6.2.ag_z_acv_go_amh_tc $56$ (not in LMFDB) 6.2.ae_j_an_i_j_ay $56$ (not in LMFDB) 6.2.ae_p_abl_dc_afl_ii $56$ (not in LMFDB) 6.2.ac_d_af_c_d_a $56$ (not in LMFDB) 6.2.ac_j_ar_bm_acl_ds $56$ (not in LMFDB) 6.2.ab_e_ae_c_ad_ai $56$ (not in LMFDB) 6.2.a_h_af_u_abd_bo $56$ (not in LMFDB) 6.2.a_h_f_u_bd_bo $56$ (not in LMFDB) 6.2.b_e_e_c_d_ai $56$ (not in LMFDB) 6.2.c_d_f_c_ad_a $56$ (not in LMFDB) 6.2.c_j_r_bm_cl_ds $56$ (not in LMFDB) 6.2.e_j_n_i_aj_ay $56$ (not in LMFDB) 6.2.e_p_bl_dc_fl_ii $56$ (not in LMFDB) 6.2.g_z_cv_go_mh_tc $56$ (not in LMFDB) 6.2.h_bc_de_hg_nn_uy $56$ (not in LMFDB) 6.2.ag_u_abv_dl_afr_ik $84$ (not in LMFDB) 6.2.af_s_abq_df_afb_hq $84$ (not in LMFDB) 6.2.ae_j_ap_ba_abx_da $84$ (not in LMFDB) 6.2.ad_c_g_aj_aj_bi $84$ (not in LMFDB) 6.2.ad_e_a_ad_ad_o $84$ (not in LMFDB) 6.2.ad_k_as_bn_acf_du $84$ (not in LMFDB) 6.2.ac_d_ah_q_ax_ba $84$ (not in LMFDB) 6.2.ac_d_d_ae_h_g $84$ (not in LMFDB) 6.2.ac_e_ah_n_ax_be $84$ (not in LMFDB) 6.2.ab_ac_c_h_ad_ak $84$ (not in LMFDB) 6.2.ab_a_a_f_ab_ag $84$ (not in LMFDB) 6.2.ab_g_ag_x_at_cc $84$ (not in LMFDB) 6.2.ab_g_ac_t_ah_by $84$ (not in LMFDB) 6.2.a_b_ad_g_aj_c $84$ (not in LMFDB) 6.2.a_b_d_g_j_c $84$ (not in LMFDB) 6.2.b_ac_ac_h_d_ak $84$ (not in LMFDB) 6.2.b_a_a_f_b_ag $84$ (not in LMFDB) 6.2.b_g_c_t_h_by $84$ (not in LMFDB) 6.2.b_g_g_x_t_cc $84$ (not in LMFDB) 6.2.c_d_ad_ae_ah_g $84$ (not in LMFDB) 6.2.c_d_h_q_x_ba $84$ (not in LMFDB) 6.2.c_e_h_n_x_be $84$ (not in LMFDB) 6.2.d_c_ag_aj_j_bi $84$ (not in LMFDB) 6.2.d_e_a_ad_d_o $84$ (not in LMFDB) 6.2.d_k_s_bn_cf_du $84$ (not in LMFDB) 6.2.e_j_p_ba_bx_da $84$ (not in LMFDB) 6.2.f_s_bq_df_fb_hq $84$ (not in LMFDB) 6.2.g_u_bv_dl_fr_ik $84$ (not in LMFDB) 6.2.af_p_abj_cn_aeb_ga $168$ (not in LMFDB) 6.2.ae_k_at_bd_abn_ca $168$ (not in LMFDB) 6.2.ae_m_abb_bz_adh_eu $168$ (not in LMFDB) 6.2.ad_h_ar_bd_abv_cy $168$ (not in LMFDB) 6.2.ad_h_ah_ab_x_abs $168$ (not in LMFDB) 6.2.ad_k_aq_bd_abf_ca $168$ (not in LMFDB) 6.2.ad_m_aw_bz_acr_eu $168$ (not in LMFDB) 6.2.ac_d_af_i_ap_y $168$ (not in LMFDB) 6.2.ac_f_aj_q_abd_bo $168$ (not in LMFDB) 6.2.ab_a_c_ab_ad_m $168$ (not in LMFDB) 6.2.ab_c_a_b_ab_e $168$ (not in LMFDB) 6.2.ab_d_ab_ab_j_am $168$ (not in LMFDB) 6.2.ab_g_ae_r_aj_bk $168$ (not in LMFDB) 6.2.ab_i_ag_bf_at_cy $168$ (not in LMFDB) 6.2.a_b_af_c_af_q $168$ (not in LMFDB) 6.2.a_b_f_c_f_q $168$ (not in LMFDB) 6.2.a_d_af_g_ap_q $168$ (not in LMFDB) 6.2.a_d_f_g_p_q $168$ (not in LMFDB) 6.2.b_a_ac_ab_d_m $168$ (not in LMFDB) 6.2.b_c_a_b_b_e $168$ (not in LMFDB) 6.2.b_d_b_ab_aj_am $168$ (not in LMFDB) 6.2.b_g_e_r_j_bk $168$ (not in LMFDB) 6.2.b_i_g_bf_t_cy $168$ (not in LMFDB) 6.2.c_d_f_i_p_y $168$ (not in LMFDB) 6.2.c_f_j_q_bd_bo $168$ (not in LMFDB) 6.2.d_h_h_ab_ax_abs $168$ (not in LMFDB) 6.2.d_h_r_bd_bv_cy $168$ (not in LMFDB) 6.2.d_k_q_bd_bf_ca $168$ (not in LMFDB) 6.2.d_m_w_bz_cr_eu $168$ (not in LMFDB) 6.2.e_k_t_bd_bn_ca $168$ (not in LMFDB) 6.2.e_m_bb_bz_dh_eu $168$ (not in LMFDB) 6.2.f_p_bj_cn_eb_ga $168$ (not in LMFDB)