# Properties

 Label 6.2.ai_bd_acg_cf_i_adc 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: $( 1 - 2 x + 2 x^{2} )^{2}( 1 - 4 x + 5 x^{2} + 2 x^{3} - 11 x^{4} + 4 x^{5} + 20 x^{6} - 32 x^{7} + 16 x^{8} )$ Frobenius angles: $\pm0.0247483856139$, $\pm0.177336015878$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.344002682545$, $\pm0.858081718947$ 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 $A(\F_{q^r})$ 1 1525 852436 48075625 1110571141 67598174800 2654320083493 248168731955625 14879912670919972 1084806584635763125

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -5 -1 19 35 35 65 65 227 415 959

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
 The isogeny class factors as 1.2.ac 2 $\times$ 4.2.ae_f_c_al 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_f_c_al : 8.0.22581504.2.
Endomorphism algebra over $\overline{\F}_{2}$
 The base change of $A$ to $\F_{2^{12}}$ is 1.4096.ey 2 $\times$ 2.4096.ahm_zkj 2 . The endomorphism algebra for each factor is: 1.4096.ey 2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$. 2.4096.ahm_zkj 2 : $\mathrm{M}_{2}($4.0.4752.1$)$
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 2 $\times$ 4.4.ag_t_abq_dd. The endomorphism algebra for each factor is: 1.4.a 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-1})$$$)$ 4.4.ag_t_abq_dd : 8.0.22581504.2.
• Endomorphism algebra over $\F_{2^{3}}$  The base change of $A$ to $\F_{2^{3}}$ is 1.8.e 2 $\times$ 4.8.c_c_ak_aep. The endomorphism algebra for each factor is: 1.8.e 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-1})$$$)$ 4.8.c_c_ak_aep : 8.0.22581504.2.
• Endomorphism algebra over $\F_{2^{4}}$  The base change of $A$ to $\F_{2^{4}}$ is 1.16.i 2 $\times$ 4.16.c_t_adq_adr. 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 $\infty$. 4.16.c_t_adq_adr : 8.0.22581504.2.
• Endomorphism algebra over $\F_{2^{6}}$  The base change of $A$ to $\F_{2^{6}}$ is 1.64.a 2 $\times$ 4.64.a_ahm_a_zkj. The endomorphism algebra for each factor is: 1.64.a 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-1})$$$)$ 4.64.a_ahm_a_zkj : 8.0.22581504.2.

## 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_f_c_ah_am_bo $2$ (not in LMFDB) 6.2.a_ad_ac_j_a_aq $2$ (not in LMFDB) 6.2.a_ad_c_j_a_aq $2$ (not in LMFDB) 6.2.e_f_ac_ah_m_bo $2$ (not in LMFDB) 6.2.i_bd_cg_cf_ai_adc $2$ (not in LMFDB) 6.2.ac_ab_i_aj_ak_bi $3$ (not in LMFDB) 6.2.ac_c_c_ad_i_ai $3$ (not in LMFDB) 6.2.ac_f_ae_j_ae_q $3$ (not in LMFDB) 6.2.e_i_o_v_ba_bi $3$ (not in LMFDB) 6.2.e_l_ba_bz_di_fa $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 6.2.ae_f_c_ah_am_bo $2$ (not in LMFDB) 6.2.a_ad_ac_j_a_aq $2$ (not in LMFDB) 6.2.a_ad_c_j_a_aq $2$ (not in LMFDB) 6.2.e_f_ac_ah_m_bo $2$ (not in LMFDB) 6.2.i_bd_cg_cf_ai_adc $2$ (not in LMFDB) 6.2.ac_ab_i_aj_ak_bi $3$ (not in LMFDB) 6.2.ac_c_c_ad_i_ai $3$ (not in LMFDB) 6.2.ac_f_ae_j_ae_q $3$ (not in LMFDB) 6.2.e_i_o_v_ba_bi $3$ (not in LMFDB) 6.2.e_l_ba_bz_di_fa $3$ (not in LMFDB) 6.2.ag_v_aca_eb_agy_km $4$ (not in LMFDB) 6.2.ac_f_ai_r_ay_bo $4$ (not in LMFDB) 6.2.ac_f_ae_j_ae_q $4$ (not in LMFDB) 6.2.c_f_e_j_e_q $4$ (not in LMFDB) 6.2.c_f_i_r_y_bo $4$ (not in LMFDB) 6.2.g_v_ca_eb_gy_km $4$ (not in LMFDB) 6.2.ag_p_au_p_ag_c $6$ (not in LMFDB) 6.2.ag_s_abi_bt_abw_ce $6$ (not in LMFDB) 6.2.ag_v_aca_eb_agy_km $6$ (not in LMFDB) 6.2.ae_i_ao_v_aba_bi $6$ (not in LMFDB) 6.2.ae_l_aba_bz_adi_fa $6$ (not in LMFDB) 6.2.ac_c_ac_f_am_q $6$ (not in LMFDB) 6.2.ac_f_ai_r_ay_bo $6$ (not in LMFDB) 6.2.a_a_ac_ad_g_c $6$ (not in LMFDB) 6.2.a_a_c_ad_ag_c $6$ (not in LMFDB) 6.2.a_d_ac_d_ag_c $6$ (not in LMFDB) 6.2.a_d_c_d_g_c $6$ (not in LMFDB) 6.2.c_ab_ai_aj_k_bi $6$ (not in LMFDB) 6.2.c_c_ac_ad_ai_ai $6$ (not in LMFDB) 6.2.c_c_c_f_m_q $6$ (not in LMFDB) 6.2.c_f_e_j_e_q $6$ (not in LMFDB) 6.2.c_f_i_r_y_bo $6$ (not in LMFDB) 6.2.g_p_u_p_g_c $6$ (not in LMFDB) 6.2.g_s_bi_bt_bw_ce $6$ (not in LMFDB) 6.2.g_v_ca_eb_gy_km $6$ (not in LMFDB) 6.2.ag_r_abc_z_ac_au $8$ (not in LMFDB) 6.2.ae_b_s_abb_au_dg $8$ (not in LMFDB) 6.2.ae_j_ao_n_ae_ae $8$ (not in LMFDB) 6.2.ae_n_abe_cj_ady_ga $8$ (not in LMFDB) 6.2.ac_b_a_ad_i_am $8$ (not in LMFDB) 6.2.ac_b_a_b_ag_m $8$ (not in LMFDB) 6.2.ac_j_aq_bl_ace_do $8$ (not in LMFDB) 6.2.a_f_ac_n_ak_bc $8$ (not in LMFDB) 6.2.a_f_c_n_k_bc $8$ (not in LMFDB) 6.2.c_b_a_ad_ai_am $8$ (not in LMFDB) 6.2.c_b_a_b_g_m $8$ (not in LMFDB) 6.2.c_j_q_bl_ce_do $8$ (not in LMFDB) 6.2.e_b_as_abb_u_dg $8$ (not in LMFDB) 6.2.e_j_o_n_e_ae $8$ (not in LMFDB) 6.2.e_n_be_cj_dy_ga $8$ (not in LMFDB) 6.2.g_r_bc_z_c_au $8$ (not in LMFDB) 6.2.ae_d_k_ar_aq_ck $24$ (not in LMFDB) 6.2.ae_e_i_at_ae_bo $24$ (not in LMFDB) 6.2.ae_h_ag_d_ai_s $24$ (not in LMFDB) 6.2.ae_k_as_z_abe_bk $24$ (not in LMFDB) 6.2.ae_m_ay_bt_acq_ea $24$ (not in LMFDB) 6.2.ac_ac_e_f_ac_ao $24$ (not in LMFDB) 6.2.ac_ac_g_ad_ae_m $24$ (not in LMFDB) 6.2.ac_a_c_b_ai_o $24$ (not in LMFDB) 6.2.ac_a_e_ad_ac_e $24$ (not in LMFDB) 6.2.ac_d_ae_h_ai_o $24$ (not in LMFDB) 6.2.ac_e_ag_j_aq_s $24$ (not in LMFDB) 6.2.ac_g_am_v_abi_by $24$ (not in LMFDB) 6.2.ac_g_ak_n_au_u $24$ (not in LMFDB) 6.2.ac_h_am_bb_abo_co $24$ (not in LMFDB) 6.2.ac_i_am_bd_abi_cq $24$ (not in LMFDB) 6.2.a_ai_a_bd_a_acq $24$ (not in LMFDB) 6.2.a_ag_a_v_a_aby $24$ (not in LMFDB) 6.2.a_ae_a_n_a_abg $24$ (not in LMFDB) 6.2.a_ac_a_f_a_ao $24$ (not in LMFDB) 6.2.a_a_a_ad_a_ae $24$ (not in LMFDB) 6.2.a_a_a_ad_a_e $24$ (not in LMFDB) 6.2.a_c_ac_b_ak_e $24$ (not in LMFDB) 6.2.a_c_a_f_a_o $24$ (not in LMFDB) 6.2.a_c_c_b_k_e $24$ (not in LMFDB) 6.2.a_e_a_n_a_bg $24$ (not in LMFDB) 6.2.a_g_a_v_a_by $24$ (not in LMFDB) 6.2.a_i_a_bd_a_cq $24$ (not in LMFDB) 6.2.c_ac_ag_ad_e_m $24$ (not in LMFDB) 6.2.c_ac_ae_f_c_ao $24$ (not in LMFDB) 6.2.c_a_ae_ad_c_e $24$ (not in LMFDB) 6.2.c_a_ac_b_i_o $24$ (not in LMFDB) 6.2.c_d_e_h_i_o $24$ (not in LMFDB) 6.2.c_e_g_j_q_s $24$ (not in LMFDB) 6.2.c_g_k_n_u_u $24$ (not in LMFDB) 6.2.c_g_m_v_bi_by $24$ (not in LMFDB) 6.2.c_h_m_bb_bo_co $24$ (not in LMFDB) 6.2.c_i_m_bd_bi_cq $24$ (not in LMFDB) 6.2.e_d_ak_ar_q_ck $24$ (not in LMFDB) 6.2.e_e_ai_at_e_bo $24$ (not in LMFDB) 6.2.e_h_g_d_i_s $24$ (not in LMFDB) 6.2.e_k_s_z_be_bk $24$ (not in LMFDB) 6.2.e_m_y_bt_cq_ea $24$ (not in LMFDB)