# Properties

 Label 6.2.aj_bp_aet_ko_asw_bcq 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: $( 1 - 2 x + 2 x^{2} )^{3}( 1 - 3 x + 5 x^{2} - 7 x^{3} + 10 x^{4} - 12 x^{5} + 8 x^{6} )$ Frobenius angles: $\pm0.105278500939$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.316838792568$, $\pm0.641249159631$ 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 11500 663494 97750000 2952713482 49596176500 2976029892586 242283150000000 16448442165424022 1275479951927687500

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -6 6 18 46 64 42 78 222 468 1126

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
 The isogeny class factors as 1.2.ac 3 $\times$ 3.2.ad_f_ah 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.ad_f_ah : 6.0.679024.1.
Endomorphism algebra over $\overline{\F}_{2}$
 The base change of $A$ to $\F_{2^{4}}$ is 1.16.i 3 $\times$ 3.16.f_br_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.f_br_fn : 6.0.679024.1.
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.b_d_af. The endomorphism algebra for each factor is: 1.4.a 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-1})$$$)$ 3.4.b_d_af : 6.0.679024.1.

## 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.af_n_ax_bm_aco_ea $2$ (not in LMFDB) 6.2.ad_f_ab_ac_c_i $2$ (not in LMFDB) 6.2.ab_b_ad_k_ak_i $2$ (not in LMFDB) 6.2.b_b_d_k_k_i $2$ (not in LMFDB) 6.2.d_f_b_ac_ac_i $2$ (not in LMFDB) 6.2.f_n_x_bm_co_ea $2$ (not in LMFDB) 6.2.j_bp_et_ko_sw_bcq $2$ (not in LMFDB) 6.2.ad_f_ad_ac_i_am $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 6.2.af_n_ax_bm_aco_ea $2$ (not in LMFDB) 6.2.ad_f_ab_ac_c_i $2$ (not in LMFDB) 6.2.ab_b_ad_k_ak_i $2$ (not in LMFDB) 6.2.b_b_d_k_k_i $2$ (not in LMFDB) 6.2.d_f_b_ac_ac_i $2$ (not in LMFDB) 6.2.f_n_x_bm_co_ea $2$ (not in LMFDB) 6.2.j_bp_et_ko_sw_bcq $2$ (not in LMFDB) 6.2.ad_f_ad_ac_i_am $3$ (not in LMFDB) 6.2.ah_z_acl_fa_aiy_ns $6$ (not in LMFDB) 6.2.ad_f_al_w_abg_bs $6$ (not in LMFDB) 6.2.ab_b_ab_c_a_ae $6$ (not in LMFDB) 6.2.b_b_b_c_a_ae $6$ (not in LMFDB) 6.2.d_f_d_ac_ai_am $6$ (not in LMFDB) 6.2.d_f_l_w_bg_bs $6$ (not in LMFDB) 6.2.h_z_cl_fa_iy_ns $6$ (not in LMFDB) 6.2.ah_bb_acv_ga_aks_qi $8$ (not in LMFDB) 6.2.af_j_ad_ao_ba_abg $8$ (not in LMFDB) 6.2.af_r_abr_dm_agc_jg $8$ (not in LMFDB) 6.2.ad_d_ab_ae_o_ay $8$ (not in LMFDB) 6.2.ad_h_an_y_abm_ce $8$ (not in LMFDB) 6.2.ad_l_az_ca_adm_fg $8$ (not in LMFDB) 6.2.ab_ad_b_g_c_aq $8$ (not in LMFDB) 6.2.ab_d_b_e_g_i $8$ (not in LMFDB) 6.2.ab_f_ah_o_aw_bg $8$ (not in LMFDB) 6.2.b_ad_ab_g_ac_aq $8$ (not in LMFDB) 6.2.b_d_ab_e_ag_i $8$ (not in LMFDB) 6.2.b_f_h_o_w_bg $8$ (not in LMFDB) 6.2.d_d_b_ae_ao_ay $8$ (not in LMFDB) 6.2.d_h_n_y_bm_ce $8$ (not in LMFDB) 6.2.d_l_z_ca_dm_fg $8$ (not in LMFDB) 6.2.f_j_d_ao_aba_abg $8$ (not in LMFDB) 6.2.f_r_br_dm_gc_jg $8$ (not in LMFDB) 6.2.h_bb_cv_ga_ks_qi $8$ (not in LMFDB) 6.2.af_l_an_m_au_bk $24$ (not in LMFDB) 6.2.af_p_abl_cy_afc_hs $24$ (not in LMFDB) 6.2.af_p_abh_cm_aei_gq $24$ (not in LMFDB) 6.2.ad_f_ah_k_am_q $24$ (not in LMFDB) 6.2.ad_j_at_bm_acm_ds $24$ (not in LMFDB) 6.2.ab_ab_ab_i_ae_ae $24$ (not in LMFDB) 6.2.ab_d_af_m_aq_u $24$ (not in LMFDB) 6.2.ab_d_ab_a_e_ai $24$ (not in LMFDB) 6.2.b_ab_b_i_e_ae $24$ (not in LMFDB) 6.2.b_d_b_a_ae_ai $24$ (not in LMFDB) 6.2.b_d_f_m_q_u $24$ (not in LMFDB) 6.2.d_f_h_k_m_q $24$ (not in LMFDB) 6.2.d_j_t_bm_cm_ds $24$ (not in LMFDB) 6.2.f_l_n_m_u_bk $24$ (not in LMFDB) 6.2.f_p_bh_cm_ei_gq $24$ (not in LMFDB) 6.2.f_p_bl_cy_fc_hs $24$ (not in LMFDB)