# Properties

 Label 6.2.aj_br_afj_na_aza_bnc 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: $( 1 - x + 2 x^{2} )( 1 - 2 x + 2 x^{2} )^{3}( 1 - 2 x + 3 x^{2} - 4 x^{3} + 4 x^{4} )$ Frobenius angles: $\pm0.174442860055$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.384973271919$, $\pm0.546783656212$ 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 $A(\F_{q^r})$ 4 28000 1906996 56000000 2671653644 93442804000 3287686987204 214570944000000 16150985219678764 1054301319263500000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -6 10 24 38 64 82 92 190 456 930

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
 The isogeny class factors as 1.2.ac 3 $\times$ 1.2.ab $\times$ 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 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-1})$$$)$ 1.2.ab : $$\Q(\sqrt{-7})$$. 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.ab $\times$ 1.16.i 3 $\times$ 2.16.ac_b. The endomorphism algebra for each factor is: 1.16.ab : $$\Q(\sqrt{-7})$$. 1.16.i 3 : $\mathrm{M}_{3}(B)$, 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 3 $\times$ 1.4.d $\times$ 2.4.c_b. The endomorphism algebra for each factor is: 1.4.a 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-1})$$$)$ 1.4.d : $$\Q(\sqrt{-7})$$. 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.ah_bb_acv_gc_ala_qy $2$ (not in LMFDB) 6.2.af_p_abf_cc_ade_eq $2$ (not in LMFDB) 6.2.af_p_abb_bi_aba_y $2$ (not in LMFDB) 6.2.ad_h_an_ba_abm_ce $2$ (not in LMFDB) 6.2.ad_h_aj_o_ao_y $2$ (not in LMFDB) 6.2.ab_d_ad_k_ak_y $2$ (not in LMFDB) 6.2.ab_d_b_g_ac_y $2$ (not in LMFDB) 6.2.b_d_ab_g_c_y $2$ (not in LMFDB) 6.2.b_d_d_k_k_y $2$ (not in LMFDB) 6.2.d_h_j_o_o_y $2$ (not in LMFDB) 6.2.d_h_n_ba_bm_ce $2$ (not in LMFDB) 6.2.f_p_bb_bi_ba_y $2$ (not in LMFDB) 6.2.f_p_bf_cc_de_eq $2$ (not in LMFDB) 6.2.h_bb_cv_gc_la_qy $2$ (not in LMFDB) 6.2.j_br_fj_na_za_bnc $2$ (not in LMFDB) 6.2.ad_h_ah_c_q_abc $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 6.2.ah_bb_acv_gc_ala_qy $2$ (not in LMFDB) 6.2.af_p_abf_cc_ade_eq $2$ (not in LMFDB) 6.2.af_p_abb_bi_aba_y $2$ (not in LMFDB) 6.2.ad_h_an_ba_abm_ce $2$ (not in LMFDB) 6.2.ad_h_aj_o_ao_y $2$ (not in LMFDB) 6.2.ab_d_ad_k_ak_y $2$ (not in LMFDB) 6.2.ab_d_b_g_ac_y $2$ (not in LMFDB) 6.2.b_d_ab_g_c_y $2$ (not in LMFDB) 6.2.b_d_d_k_k_y $2$ (not in LMFDB) 6.2.d_h_j_o_o_y $2$ (not in LMFDB) 6.2.d_h_n_ba_bm_ce $2$ (not in LMFDB) 6.2.f_p_bb_bi_ba_y $2$ (not in LMFDB) 6.2.f_p_bf_cc_de_eq $2$ (not in LMFDB) 6.2.h_bb_cv_gc_la_qy $2$ (not in LMFDB) 6.2.j_br_fj_na_za_bnc $2$ (not in LMFDB) 6.2.ad_h_ah_c_q_abc $3$ (not in LMFDB) 6.2.ah_bb_acx_gk_als_sa $6$ (not in LMFDB) 6.2.af_p_abl_da_afg_hw $6$ (not in LMFDB) 6.2.ad_h_ap_ba_abo_ci $6$ (not in LMFDB) 6.2.ad_h_al_o_aq_u $6$ (not in LMFDB) 6.2.ab_d_aj_k_aq_bk $6$ (not in LMFDB) 6.2.ab_d_af_g_ai_m $6$ (not in LMFDB) 6.2.ab_d_ab_c_i_ae $6$ (not in LMFDB) 6.2.b_d_b_c_ai_ae $6$ (not in LMFDB) 6.2.b_d_f_g_i_m $6$ (not in LMFDB) 6.2.b_d_j_k_q_bk $6$ (not in LMFDB) 6.2.d_h_h_c_aq_abc $6$ (not in LMFDB) 6.2.d_h_l_o_q_u $6$ (not in LMFDB) 6.2.d_h_p_ba_bo_ci $6$ (not in LMFDB) 6.2.f_p_bl_da_fg_hw $6$ (not in LMFDB) 6.2.h_bb_cx_gk_ls_sa $6$ (not in LMFDB) 6.2.ah_bd_adh_ho_aoc_vw $8$ (not in LMFDB) 6.2.af_l_al_ag_bq_adc $8$ (not in LMFDB) 6.2.af_r_abr_do_agg_jo $8$ (not in LMFDB) 6.2.af_t_abz_ek_ahy_mi $8$ (not in LMFDB) 6.2.ad_d_ab_ac_o_abg $8$ (not in LMFDB) 6.2.ad_f_af_ae_w_abo $8$ (not in LMFDB) 6.2.ad_j_ar_bg_abu_cu $8$ (not in LMFDB) 6.2.ad_j_an_u_ao_y $8$ (not in LMFDB) 6.2.ad_l_az_cc_adm_fo $8$ (not in LMFDB) 6.2.ad_n_abd_cq_aek_hc $8$ (not in LMFDB) 6.2.ab_ab_b_ac_c_a $8$ (not in LMFDB) 6.2.ab_ab_f_ag_ag_q $8$ (not in LMFDB) 6.2.ab_b_ad_ae_k_ai $8$ (not in LMFDB) 6.2.ab_f_ah_q_as_bo $8$ (not in LMFDB) 6.2.ab_f_ad_m_ac_y $8$ (not in LMFDB) 6.2.ab_h_ah_w_aw_bw $8$ (not in LMFDB) 6.2.ab_h_ad_s_c_bg $8$ (not in LMFDB) 6.2.ab_j_al_bk_abu_dk $8$ (not in LMFDB) 6.2.b_ab_af_ag_g_q $8$ (not in LMFDB) 6.2.b_ab_ab_ac_ac_a $8$ (not in LMFDB) 6.2.b_b_d_ae_ak_ai $8$ (not in LMFDB) 6.2.b_f_d_m_c_y $8$ (not in LMFDB) 6.2.b_f_h_q_s_bo $8$ (not in LMFDB) 6.2.b_h_d_s_ac_bg $8$ (not in LMFDB) 6.2.b_h_h_w_w_bw $8$ (not in LMFDB) 6.2.b_j_l_bk_bu_dk $8$ (not in LMFDB) 6.2.d_d_b_ac_ao_abg $8$ (not in LMFDB) 6.2.d_f_f_ae_aw_abo $8$ (not in LMFDB) 6.2.d_j_n_u_o_y $8$ (not in LMFDB) 6.2.d_j_r_bg_bu_cu $8$ (not in LMFDB) 6.2.d_l_z_cc_dm_fo $8$ (not in LMFDB) 6.2.d_n_bd_cq_ek_hc $8$ (not in LMFDB) 6.2.f_l_l_ag_abq_adc $8$ (not in LMFDB) 6.2.f_r_br_do_gg_jo $8$ (not in LMFDB) 6.2.f_t_bz_ek_hy_mi $8$ (not in LMFDB) 6.2.h_bd_dh_ho_oc_vw $8$ (not in LMFDB) 6.2.b_d_b_c_ai_ae $12$ (not in LMFDB) 6.2.af_n_av_y_au_u $24$ (not in LMFDB) 6.2.af_r_abt_ds_agq_ke $24$ (not in LMFDB) 6.2.af_r_abp_dg_afo_im $24$ (not in LMFDB) 6.2.ad_f_ah_m_am_m $24$ (not in LMFDB) 6.2.ad_h_al_o_am_q $24$ (not in LMFDB) 6.2.ad_j_ax_bs_acy_eq $24$ (not in LMFDB) 6.2.ad_j_at_bo_acm_dw $24$ (not in LMFDB) 6.2.ad_l_ax_by_adc_ey $24$ (not in LMFDB) 6.2.ab_b_ab_e_ae_m $24$ (not in LMFDB) 6.2.ab_b_d_a_ae_u $24$ (not in LMFDB) 6.2.ab_d_af_g_ae_q $24$ (not in LMFDB) 6.2.ab_f_af_i_am_i $24$ (not in LMFDB) 6.2.ab_f_af_q_aq_bk $24$ (not in LMFDB) 6.2.ab_f_ab_e_m_ai $24$ (not in LMFDB) 6.2.ab_f_ab_m_a_bc $24$ (not in LMFDB) 6.2.ab_h_aj_ba_abg_cm $24$ (not in LMFDB) 6.2.b_b_ad_a_e_u $24$ (not in LMFDB) 6.2.b_b_b_e_e_m $24$ (not in LMFDB) 6.2.b_d_f_g_e_q $24$ (not in LMFDB) 6.2.b_f_b_e_am_ai $24$ (not in LMFDB) 6.2.b_f_b_m_a_bc $24$ (not in LMFDB) 6.2.b_f_f_i_m_i $24$ (not in LMFDB) 6.2.b_f_f_q_q_bk $24$ (not in LMFDB) 6.2.b_h_j_ba_bg_cm $24$ (not in LMFDB) 6.2.d_f_h_m_m_m $24$ (not in LMFDB) 6.2.d_h_l_o_m_q $24$ (not in LMFDB) 6.2.d_j_t_bo_cm_dw $24$ (not in LMFDB) 6.2.d_j_x_bs_cy_eq $24$ (not in LMFDB) 6.2.d_l_x_by_dc_ey $24$ (not in LMFDB) 6.2.f_n_v_y_u_u $24$ (not in LMFDB) 6.2.f_r_bp_dg_fo_im $24$ (not in LMFDB) 6.2.f_r_bt_ds_gq_ke $24$ (not in LMFDB)