# Properties

 Label 6.2.ai_bg_adg_gj_aki_pc Base Field $\F_{2}$ Dimension $6$ Ordinary No $p$-rank $4$ Principally polarizable Yes Contains a Jacobian No

## Invariants

 Base field: $\F_{2}$ Dimension: $6$ L-polynomial: $( 1 - 2 x + 2 x^{2} )^{2}( 1 - 4 x + 8 x^{2} - 12 x^{3} + 17 x^{4} - 24 x^{5} + 32 x^{6} - 32 x^{7} + 16 x^{8} )$ Frobenius angles: $\pm0.0755571399449$, $\pm0.203216343788$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.424442860055$, $\pm0.703216343788$ Angle rank: $2$ (numerical)

This isogeny class is not simple.

## Newton polygon

 $p$-rank: $4$ Slopes: $[0, 0, 0, 0, 1/2, 1/2, 1/2, 1/2, 1, 1, 1, 1]$

## 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 $A(\F_{q^r})$ 2 7300 405938 53290000 1563467842 72602011300 5146903416146 216584202240000 13706001241610834 1152887506347182500

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -5 5 13 37 45 65 149 189 373 1025

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
 The isogeny class factors as 1.2.ac 2 $\times$ 4.2.ae_i_am_r 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 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-1})$$$)$ 4.2.ae_i_am_r : 8.0.18939904.2.
Endomorphism algebra over $\overline{\F}_{2}$
 The base change of $A$ to $\F_{2^{4}}$ is 1.16.i 2 $\times$ 2.16.c_b 2 . The endomorphism algebra for each factor is: 1.16.i 2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$. 2.16.c_b 2 : $\mathrm{M}_{2}($4.0.1088.2$)$
All geometric endomorphisms are defined over $\F_{2^{4}}$.
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 2 $\times$ 4.4.a_c_a_b. The endomorphism algebra for each factor is: 1.4.a 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-1})$$$)$ 4.4.a_c_a_b : 8.0.18939904.2.

## 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_i_am_v_abo_cm $2$ (not in LMFDB) 6.2.a_a_ae_f_ae_i $2$ (not in LMFDB) 6.2.a_a_e_f_e_i $2$ (not in LMFDB) 6.2.e_i_m_v_bo_cm $2$ (not in LMFDB) 6.2.i_bg_dg_gj_ki_pc $2$ (not in LMFDB) 6.2.ac_c_a_ad_c_c $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 6.2.ae_i_am_v_abo_cm $2$ (not in LMFDB) 6.2.a_a_ae_f_ae_i $2$ (not in LMFDB) 6.2.a_a_e_f_e_i $2$ (not in LMFDB) 6.2.e_i_m_v_bo_cm $2$ (not in LMFDB) 6.2.i_bg_dg_gj_ki_pc $2$ (not in LMFDB) 6.2.ac_c_a_ad_c_c $3$ (not in LMFDB) 6.2.ag_s_abo_cz_afa_hm $6$ (not in LMFDB) 6.2.c_c_a_ad_ac_c $6$ (not in LMFDB) 6.2.g_s_bo_cz_fa_hm $6$ (not in LMFDB) 6.2.ai_bi_adw_iv_aqm_zo $8$ (not in LMFDB) 6.2.ag_u_abw_dp_afy_iu $8$ (not in LMFDB) 6.2.ag_w_aci_fd_aji_oi $8$ (not in LMFDB) 6.2.ae_e_e_al_i_ae $8$ (not in LMFDB) 6.2.ae_g_ae_ad_y_aca $8$ (not in LMFDB) 6.2.ae_g_a_al_m_ai $8$ (not in LMFDB) 6.2.ae_k_au_bl_ace_dc $8$ (not in LMFDB) 6.2.ae_k_aq_v_au_y $8$ (not in LMFDB) 6.2.ae_m_abc_cb_adk_fc $8$ (not in LMFDB) 6.2.ae_o_abk_cz_afg_ie $8$ (not in LMFDB) 6.2.ac_c_a_ad_g_am $8$ (not in LMFDB) 6.2.ac_e_ai_n_aw_bk $8$ (not in LMFDB) 6.2.ac_g_am_v_abe_ca $8$ (not in LMFDB) 6.2.ac_g_ai_n_ak_u $8$ (not in LMFDB) 6.2.a_ag_a_n_a_au $8$ (not in LMFDB) 6.2.a_ac_a_ad_a_m $8$ (not in LMFDB) 6.2.a_ac_a_f_a_aq $8$ (not in LMFDB) 6.2.a_c_ae_f_ae_y $8$ (not in LMFDB) 6.2.a_c_a_ad_a_am $8$ (not in LMFDB) 6.2.a_c_a_f_a_q $8$ (not in LMFDB) 6.2.a_c_e_f_e_y $8$ (not in LMFDB) 6.2.a_g_a_n_a_u $8$ (not in LMFDB) 6.2.c_c_a_ad_ag_am $8$ (not in LMFDB) 6.2.c_e_i_n_w_bk $8$ (not in LMFDB) 6.2.c_g_i_n_k_u $8$ (not in LMFDB) 6.2.c_g_m_v_be_ca $8$ (not in LMFDB) 6.2.e_e_ae_al_ai_ae $8$ (not in LMFDB) 6.2.e_g_a_al_am_ai $8$ (not in LMFDB) 6.2.e_g_e_ad_ay_aca $8$ (not in LMFDB) 6.2.e_k_q_v_u_y $8$ (not in LMFDB) 6.2.e_k_u_bl_ce_dc $8$ (not in LMFDB) 6.2.e_m_bc_cb_dk_fc $8$ (not in LMFDB) 6.2.e_o_bk_cz_fg_ie $8$ (not in LMFDB) 6.2.g_u_bw_dp_fy_iu $8$ (not in LMFDB) 6.2.g_w_ci_fd_ji_oi $8$ (not in LMFDB) 6.2.i_bi_dw_iv_qm_zo $8$ (not in LMFDB) 6.2.ag_r_aba_r_q_abw $24$ (not in LMFDB) 6.2.ag_u_aca_ej_ahu_lu $24$ (not in LMFDB) 6.2.ae_g_ae_f_aq_be $24$ (not in LMFDB) 6.2.ae_h_ai_h_c_ao $24$ (not in LMFDB) 6.2.ae_i_am_r_aq_o $24$ (not in LMFDB) 6.2.ae_j_am_j_g_au $24$ (not in LMFDB) 6.2.ae_k_au_bl_acm_du $24$ (not in LMFDB) 6.2.ae_m_abc_cf_ads_fq $24$ (not in LMFDB) 6.2.ac_ad_k_ad_am_u $24$ (not in LMFDB) 6.2.ac_ab_g_ab_ai_o $24$ (not in LMFDB) 6.2.ac_a_a_b_g_ao $24$ (not in LMFDB) 6.2.ac_b_c_b_ai_q $24$ (not in LMFDB) 6.2.ac_b_c_b_ae_i $24$ (not in LMFDB) 6.2.ac_d_ac_d_a_c $24$ (not in LMFDB) 6.2.ac_e_ai_j_ak_s $24$ (not in LMFDB) 6.2.ac_e_ae_b_k_ao $24$ (not in LMFDB) 6.2.ac_f_ag_f_e_ae $24$ (not in LMFDB) 6.2.a_ae_a_j_a_as $24$ (not in LMFDB) 6.2.a_ab_ae_ab_c_s $24$ (not in LMFDB) 6.2.a_ab_e_ab_ac_s $24$ (not in LMFDB) 6.2.a_a_a_b_a_ao $24$ (not in LMFDB) 6.2.a_a_a_b_a_o $24$ (not in LMFDB) 6.2.a_b_a_b_ac_m $24$ (not in LMFDB) 6.2.a_b_a_b_c_m $24$ (not in LMFDB) 6.2.a_e_a_j_a_s $24$ (not in LMFDB) 6.2.c_ad_ak_ad_m_u $24$ (not in LMFDB) 6.2.c_ab_ag_ab_i_o $24$ (not in LMFDB) 6.2.c_a_a_b_ag_ao $24$ (not in LMFDB) 6.2.c_b_ac_b_e_i $24$ (not in LMFDB) 6.2.c_b_ac_b_i_q $24$ (not in LMFDB) 6.2.c_d_c_d_a_c $24$ (not in LMFDB) 6.2.c_e_e_b_ak_ao $24$ (not in LMFDB) 6.2.c_e_i_j_k_s $24$ (not in LMFDB) 6.2.c_f_g_f_ae_ae $24$ (not in LMFDB) 6.2.e_g_e_f_q_be $24$ (not in LMFDB) 6.2.e_h_i_h_ac_ao $24$ (not in LMFDB) 6.2.e_i_m_r_q_o $24$ (not in LMFDB) 6.2.e_j_m_j_ag_au $24$ (not in LMFDB) 6.2.e_k_u_bl_cm_du $24$ (not in LMFDB) 6.2.e_m_bc_cf_ds_fq $24$ (not in LMFDB) 6.2.g_r_ba_r_aq_abw $24$ (not in LMFDB) 6.2.g_u_ca_ej_hu_lu $24$ (not in LMFDB)