# Properties

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

## Invariants

 Base field: $\F_{2}$ Dimension: $6$ L-polynomial: $( 1 - 2 x + 2 x^{2} )^{4}( 1 - 2 x + 3 x^{2} - 4 x^{3} + 4 x^{4} )$ Frobenius angles: $\pm0.174442860055$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.546783656212$ Angle rank: $2$ (numerical)

This isogeny class is not simple.

## Newton polygon $p$-rank: $2$ Slopes: $[0, 0, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1, 1]$

## 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 17500 1770782 87500000 4978990882 108460397500 2616257954606 167633550000000 14997343418273138 1116383111823437500

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -7 7 23 47 83 91 63 127 419 987

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
 The isogeny class factors as 1.2.ac 4 $\times$ 2.2.ac_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 4 : $\mathrm{M}_{4}($$$\Q(\sqrt{-1})$$$)$ 2.2.ac_d : 4.0.1088.2.
Endomorphism algebra over $\overline{\F}_{2}$
 The base change of $A$ to $\F_{2^{4}}$ is 1.16.i 4 $\times$ 2.16.ac_b. The endomorphism algebra for each factor is: 1.16.i 4 : $\mathrm{M}_{4}(B)$, where $B$ is the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$. 2.16.ac_b : 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 4 $\times$ 2.4.c_b. The endomorphism algebra for each factor is: 1.4.a 4 : $\mathrm{M}_{4}($$$\Q(\sqrt{-1})$$$)$ 2.4.c_b : 4.0.1088.2.

## 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.ag_t_abo_cq_aea_fw $2$ (not in LMFDB) 6.2.ag_t_abk_bs_abg_y $2$ (not in LMFDB) 6.2.ac_d_ae_m_aq_y $2$ (not in LMFDB) 6.2.ac_d_a_e_ai_y $2$ (not in LMFDB) 6.2.c_d_a_e_i_y $2$ (not in LMFDB) 6.2.c_d_e_m_q_y $2$ (not in LMFDB) 6.2.g_t_bk_bs_bg_y $2$ (not in LMFDB) 6.2.g_t_bo_cq_ea_fw $2$ (not in LMFDB) 6.2.k_bz_gq_qm_bga_bye $2$ (not in LMFDB) 6.2.ae_j_ak_c_u_abo $3$ (not in LMFDB) 6.2.c_d_i_i_i_u $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 6.2.ag_t_abo_cq_aea_fw $2$ (not in LMFDB) 6.2.ag_t_abk_bs_abg_y $2$ (not in LMFDB) 6.2.ac_d_ae_m_aq_y $2$ (not in LMFDB) 6.2.ac_d_a_e_ai_y $2$ (not in LMFDB) 6.2.c_d_a_e_i_y $2$ (not in LMFDB) 6.2.c_d_e_m_q_y $2$ (not in LMFDB) 6.2.g_t_bk_bs_bg_y $2$ (not in LMFDB) 6.2.g_t_bo_cq_ea_fw $2$ (not in LMFDB) 6.2.k_bz_gq_qm_bga_bye $2$ (not in LMFDB) 6.2.ae_j_ak_c_u_abo $3$ (not in LMFDB) 6.2.c_d_i_i_i_u $3$ (not in LMFDB) 6.2.a_b_ac_ac_i_e $5$ (not in LMFDB) 6.2.ai_bh_adq_ic_aoy_xc $6$ (not in LMFDB) 6.2.ag_t_abw_ea_ahc_kq $6$ (not in LMFDB) 6.2.ae_j_as_bi_aca_cu $6$ (not in LMFDB) 6.2.ae_j_ao_s_au_y $6$ (not in LMFDB) 6.2.ac_d_ai_i_ai_u $6$ (not in LMFDB) 6.2.ac_d_ae_a_i_am $6$ (not in LMFDB) 6.2.a_b_ag_c_ae_y $6$ (not in LMFDB) 6.2.a_b_ac_c_ae_i $6$ (not in LMFDB) 6.2.a_b_c_c_e_i $6$ (not in LMFDB) 6.2.a_b_g_c_e_y $6$ (not in LMFDB) 6.2.c_d_e_a_ai_am $6$ (not in LMFDB) 6.2.e_j_k_c_au_abo $6$ (not in LMFDB) 6.2.e_j_o_s_u_y $6$ (not in LMFDB) 6.2.e_j_s_bi_ca_cu $6$ (not in LMFDB) 6.2.g_t_bw_ea_hc_kq $6$ (not in LMFDB) 6.2.i_bh_dq_ic_oy_xc $6$ (not in LMFDB) 6.2.ai_bj_aec_jo_asa_bca $8$ (not in LMFDB) 6.2.ag_p_aq_ai_ce_aea $8$ (not in LMFDB) 6.2.ag_x_acm_fo_ake_ps $8$ (not in LMFDB) 6.2.ae_h_ag_ae_bc_ace $8$ (not in LMFDB) 6.2.ae_l_aw_bo_aci_dk $8$ (not in LMFDB) 6.2.ae_l_as_y_au_y $8$ (not in LMFDB) 6.2.ae_p_abm_dg_afs_iy $8$ (not in LMFDB) 6.2.ac_af_m_e_aq_i $8$ (not in LMFDB) 6.2.ac_ab_e_a_a_ai $8$ (not in LMFDB) 6.2.ac_ab_i_ai_ai_y $8$ (not in LMFDB) 6.2.ac_d_ae_ae_q_ay $8$ (not in LMFDB) 6.2.ac_h_am_y_abg_ce $8$ (not in LMFDB) 6.2.ac_h_ai_q_ai_y $8$ (not in LMFDB) 6.2.ac_l_au_ca_adc_fg $8$ (not in LMFDB) 6.2.a_ab_ac_ae_e_i $8$ (not in LMFDB) 6.2.a_ab_c_ae_ae_i $8$ (not in LMFDB) 6.2.a_d_ac_i_ae_y $8$ (not in LMFDB) 6.2.a_d_c_i_e_y $8$ (not in LMFDB) 6.2.a_h_ac_u_am_bo $8$ (not in LMFDB) 6.2.a_h_c_u_m_bo $8$ (not in LMFDB) 6.2.c_af_am_e_q_i $8$ (not in LMFDB) 6.2.c_ab_ai_ai_i_y $8$ (not in LMFDB) 6.2.c_ab_ae_a_a_ai $8$ (not in LMFDB) 6.2.c_d_e_ae_aq_ay $8$ (not in LMFDB) 6.2.c_h_i_q_i_y $8$ (not in LMFDB) 6.2.c_h_m_y_bg_ce $8$ (not in LMFDB) 6.2.c_l_u_ca_dc_fg $8$ (not in LMFDB) 6.2.e_h_g_ae_abc_ace $8$ (not in LMFDB) 6.2.e_l_s_y_u_y $8$ (not in LMFDB) 6.2.e_l_w_bo_ci_dk $8$ (not in LMFDB) 6.2.e_p_bm_dg_fs_iy $8$ (not in LMFDB) 6.2.g_p_q_ai_ace_aea $8$ (not in LMFDB) 6.2.g_x_cm_fo_ke_ps $8$ (not in LMFDB) 6.2.i_bj_ec_jo_sa_bca $8$ (not in LMFDB) 6.2.ae_j_ao_o_ai_e $10$ (not in LMFDB) 6.2.a_b_c_ac_ai_e $10$ (not in LMFDB) 6.2.e_j_o_o_i_e $10$ (not in LMFDB) 6.2.ac_d_ae_e_a_a $16$ (not in LMFDB) 6.2.c_d_e_e_a_a $16$ (not in LMFDB) 6.2.ag_r_abc_be_ay_y $24$ (not in LMFDB) 6.2.ag_v_ace_es_aim_my $24$ (not in LMFDB) 6.2.ag_v_aca_ec_ahc_ku $24$ (not in LMFDB) 6.2.ae_f_ac_ac_u_abw $24$ (not in LMFDB) 6.2.ae_h_ak_q_aq_m $24$ (not in LMFDB) 6.2.ae_j_ao_s_aq_q $24$ (not in LMFDB) 6.2.ae_l_aba_ca_adk_fc $24$ (not in LMFDB) 6.2.ae_n_abi_cs_aeu_hk $24$ (not in LMFDB) 6.2.ae_n_abe_ck_aea_ge $24$ (not in LMFDB) 6.2.ac_ad_i_c_ai_a $24$ (not in LMFDB) 6.2.ac_ab_e_e_ai_e $24$ (not in LMFDB) 6.2.ac_b_a_ac_i_aq $24$ (not in LMFDB) 6.2.ac_b_a_g_ai_i $24$ (not in LMFDB) 6.2.ac_b_e_ac_ai_y $24$ (not in LMFDB) 6.2.ac_d_ae_i_ai_m $24$ (not in LMFDB) 6.2.ac_f_am_s_abc_bw $24$ (not in LMFDB) 6.2.ac_f_ai_k_am_q $24$ (not in LMFDB) 6.2.ac_f_ai_k_ai_q $24$ (not in LMFDB) 6.2.ac_f_ai_s_ay_bo $24$ (not in LMFDB) 6.2.ac_f_ae_c_m_aq $24$ (not in LMFDB) 6.2.ac_f_ae_k_ai_y $24$ (not in LMFDB) 6.2.ac_h_am_bc_abo_cq $24$ (not in LMFDB) 6.2.ac_j_aq_bm_ace_ds $24$ (not in LMFDB) 6.2.a_ad_ac_ac_e_q $24$ (not in LMFDB) 6.2.a_ad_c_ac_ae_q $24$ (not in LMFDB) 6.2.a_ab_ac_a_a_m $24$ (not in LMFDB) 6.2.a_ab_c_a_a_m $24$ (not in LMFDB) 6.2.a_b_ac_c_a_q $24$ (not in LMFDB) 6.2.a_b_c_c_a_q $24$ (not in LMFDB) 6.2.a_d_ac_e_ai_e $24$ (not in LMFDB) 6.2.a_d_c_e_i_e $24$ (not in LMFDB) 6.2.a_f_ac_g_am_a $24$ (not in LMFDB) 6.2.a_f_ac_o_ai_bg $24$ (not in LMFDB) 6.2.a_f_c_g_m_a $24$ (not in LMFDB) 6.2.a_f_c_o_i_bg $24$ (not in LMFDB) 6.2.c_ad_ai_c_i_a $24$ (not in LMFDB) 6.2.c_ab_ae_e_i_e $24$ (not in LMFDB) 6.2.c_b_ae_ac_i_y $24$ (not in LMFDB) 6.2.c_b_a_ac_ai_aq $24$ (not in LMFDB) 6.2.c_b_a_g_i_i $24$ (not in LMFDB) 6.2.c_d_e_i_i_m $24$ (not in LMFDB) 6.2.c_f_e_c_am_aq $24$ (not in LMFDB) 6.2.c_f_e_k_i_y $24$ (not in LMFDB) 6.2.c_f_i_k_i_q $24$ (not in LMFDB) 6.2.c_f_i_k_m_q $24$ (not in LMFDB) 6.2.c_f_i_s_y_bo $24$ (not in LMFDB) 6.2.c_f_m_s_bc_bw $24$ (not in LMFDB) 6.2.c_h_m_bc_bo_cq $24$ (not in LMFDB) 6.2.c_j_q_bm_ce_ds $24$ (not in LMFDB) 6.2.e_f_c_ac_au_abw $24$ (not in LMFDB) 6.2.e_h_k_q_q_m $24$ (not in LMFDB) 6.2.e_j_o_s_q_q $24$ (not in LMFDB) 6.2.e_l_ba_ca_dk_fc $24$ (not in LMFDB) 6.2.e_n_be_ck_ea_ge $24$ (not in LMFDB) 6.2.e_n_bi_cs_eu_hk $24$ (not in LMFDB) 6.2.g_r_bc_be_y_y $24$ (not in LMFDB) 6.2.g_v_ca_ec_hc_ku $24$ (not in LMFDB) 6.2.g_v_ce_es_im_my $24$ (not in LMFDB) 6.2.ac_b_a_c_a_ae $40$ (not in LMFDB) 6.2.ac_f_ai_o_aq_bc $40$ (not in LMFDB) 6.2.c_b_a_c_a_ae $40$ (not in LMFDB) 6.2.c_f_i_o_q_bc $40$ (not in LMFDB)