# Properties

 Label 6.2.ai_be_acn_de_aby_q 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: $( 1 - 2 x + 2 x^{2} )^{3}( 1 - 2 x + 3 x^{3} - 8 x^{5} + 8 x^{6} )$ Frobenius angles: $\pm0.0889496890695$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.297004294965$, $\pm0.823081333977$ Angle rank: $3$ (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 4000 1388504 109000000 1608753982 77756224000 2388274457018 204850458000000 17256562962067208 1243164639590500000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -5 1 22 49 45 70 51 177 490 1101

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
 The isogeny class factors as 1.2.ac 3 $\times$ 3.2.ac_a_d 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}($$$\Q(\sqrt{-1})$$$)$ 3.2.ac_a_d : 6.0.1539727.2.
Endomorphism algebra over $\overline{\F}_{2}$
 The base change of $A$ to $\F_{2^{4}}$ is 1.16.i 3 $\times$ 3.16.i_bo_fn. The endomorphism algebra for each factor is: 1.16.i 3 : $\mathrm{M}_{3}(B)$, where $B$ is the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$. 3.16.i_bo_fn : 6.0.1539727.2.
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 3 $\times$ 3.4.ae_m_az. The endomorphism algebra for each factor is: 1.4.a 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-1})$$$)$ 3.4.ae_m_az : 6.0.1539727.2.

## 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_g_ab_ac_as_bw $2$ (not in LMFDB) 6.2.ae_g_b_ak_c_q $2$ (not in LMFDB) 6.2.a_ac_ab_k_ac_aq $2$ (not in LMFDB) 6.2.a_ac_b_k_c_aq $2$ (not in LMFDB) 6.2.e_g_ab_ak_ac_q $2$ (not in LMFDB) 6.2.e_g_b_ac_s_bw $2$ (not in LMFDB) 6.2.i_be_cn_de_by_q $2$ (not in LMFDB) 6.2.ac_a_h_ai_ai_bc $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 6.2.ae_g_ab_ac_as_bw $2$ (not in LMFDB) 6.2.ae_g_b_ak_c_q $2$ (not in LMFDB) 6.2.a_ac_ab_k_ac_aq $2$ (not in LMFDB) 6.2.a_ac_b_k_c_aq $2$ (not in LMFDB) 6.2.e_g_ab_ak_ac_q $2$ (not in LMFDB) 6.2.e_g_b_ac_s_bw $2$ (not in LMFDB) 6.2.i_be_cn_de_by_q $2$ (not in LMFDB) 6.2.ac_a_h_ai_ai_bc $3$ (not in LMFDB) 6.2.a_ac_b_k_c_aq $4$ (not in LMFDB) 6.2.ag_q_az_bc_abg_bs $6$ (not in LMFDB) 6.2.ac_a_ab_i_ai_e $6$ (not in LMFDB) 6.2.ac_a_b_e_a_am $6$ (not in LMFDB) 6.2.c_a_ah_ai_i_bc $6$ (not in LMFDB) 6.2.c_a_ab_e_a_am $6$ (not in LMFDB) 6.2.c_a_b_i_i_e $6$ (not in LMFDB) 6.2.g_q_z_bc_bg_bs $6$ (not in LMFDB) 6.2.ag_s_abh_bo_abi_bg $8$ (not in LMFDB) 6.2.ae_c_p_aba_ao_cu $8$ (not in LMFDB) 6.2.ae_k_ar_w_aw_y $8$ (not in LMFDB) 6.2.ac_ac_h_ae_ag_q $8$ (not in LMFDB) 6.2.ac_c_ab_e_ak_q $8$ (not in LMFDB) 6.2.ac_c_b_a_c_a $8$ (not in LMFDB) 6.2.ac_g_aj_m_ao_q $8$ (not in LMFDB) 6.2.a_ag_ab_s_c_abo $8$ (not in LMFDB) 6.2.a_ag_b_s_ac_abo $8$ (not in LMFDB) 6.2.a_c_ab_c_ag_i $8$ (not in LMFDB) 6.2.a_c_b_c_g_i $8$ (not in LMFDB) 6.2.c_ac_ah_ae_g_q $8$ (not in LMFDB) 6.2.c_c_ab_a_ac_a $8$ (not in LMFDB) 6.2.c_c_b_e_k_q $8$ (not in LMFDB) 6.2.c_g_j_m_o_q $8$ (not in LMFDB) 6.2.e_c_ap_aba_o_cu $8$ (not in LMFDB) 6.2.e_k_r_w_w_y $8$ (not in LMFDB) 6.2.g_s_bh_bo_bi_bg $8$ (not in LMFDB) 6.2.ae_e_h_ao_aq_ci $24$ (not in LMFDB) 6.2.ae_i_an_s_au_y $24$ (not in LMFDB) 6.2.ae_i_aj_k_au_bk $24$ (not in LMFDB) 6.2.ac_a_d_a_ai_q $24$ (not in LMFDB) 6.2.ac_e_af_i_am_q $24$ (not in LMFDB) 6.2.a_ae_ab_o_a_abc $24$ (not in LMFDB) 6.2.a_ae_b_o_a_abc $24$ (not in LMFDB) 6.2.a_a_ad_ac_e_i $24$ (not in LMFDB) 6.2.a_a_ab_g_ae_ae $24$ (not in LMFDB) 6.2.a_a_b_g_e_ae $24$ (not in LMFDB) 6.2.a_a_d_ac_ae_i $24$ (not in LMFDB) 6.2.c_a_ad_a_i_q $24$ (not in LMFDB) 6.2.c_e_f_i_m_q $24$ (not in LMFDB) 6.2.e_e_ah_ao_q_ci $24$ (not in LMFDB) 6.2.e_i_j_k_u_bk $24$ (not in LMFDB) 6.2.e_i_n_s_u_y $24$ (not in LMFDB)