# Properties

 Label 6.2.aj_bq_afb_lu_avy_bhw 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: $( 1 - 2 x + 2 x^{2} )^{3}( 1 - 3 x + 6 x^{2} - 9 x^{3} + 12 x^{4} - 12 x^{5} + 8 x^{6} )$ Frobenius angles: $\pm0.147012170705$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.341962716420$, $\pm0.600633654388$ 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 $A(\F_{q^r})$ 3 19125 1127061 88453125 3233980083 64665124875 2947700052888 255414501703125 17508803066335269 1110720565694109375

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -6 8 21 44 69 59 78 236 498 983

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
 The isogeny class factors as 1.2.ac 3 $\times$ 3.2.ad_g_aj 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_g_aj : 6.0.465831.1.
Endomorphism algebra over $\overline{\F}_{2}$
 The base change of $A$ to $\F_{2^{4}}$ is 1.16.i 3 $\times$ 3.16.d_bq_db. 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.d_bq_db : 6.0.465831.1.
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.d_g_h. The endomorphism algebra for each factor is: 1.4.a 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-1})$$$)$ 3.4.d_g_h : 6.0.465831.1.

## 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_o_abb_bu_acw_ei $2$ (not in LMFDB) 6.2.ad_g_af_g_ag_q $2$ (not in LMFDB) 6.2.ab_c_ad_k_ak_q $2$ (not in LMFDB) 6.2.b_c_d_k_k_q $2$ (not in LMFDB) 6.2.d_g_f_g_g_q $2$ (not in LMFDB) 6.2.f_o_bb_bu_cw_ei $2$ (not in LMFDB) 6.2.j_bq_fb_lu_vy_bhw $2$ (not in LMFDB) 6.2.ad_g_af_a_m_au $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 6.2.af_o_abb_bu_acw_ei $2$ (not in LMFDB) 6.2.ad_g_af_g_ag_q $2$ (not in LMFDB) 6.2.ab_c_ad_k_ak_q $2$ (not in LMFDB) 6.2.b_c_d_k_k_q $2$ (not in LMFDB) 6.2.d_g_f_g_g_q $2$ (not in LMFDB) 6.2.f_o_bb_bu_cw_ei $2$ (not in LMFDB) 6.2.j_bq_fb_lu_vy_bhw $2$ (not in LMFDB) 6.2.ad_g_af_a_m_au $3$ (not in LMFDB) 6.2.ah_ba_acr_fs_aki_pw $6$ (not in LMFDB) 6.2.ad_g_an_y_abk_ca $6$ (not in LMFDB) 6.2.ab_c_ad_e_ae_e $6$ (not in LMFDB) 6.2.b_c_d_e_e_e $6$ (not in LMFDB) 6.2.d_g_f_a_am_au $6$ (not in LMFDB) 6.2.d_g_n_y_bk_ca $6$ (not in LMFDB) 6.2.h_ba_cr_fs_ki_pw $6$ (not in LMFDB) 6.2.ah_bc_adb_gu_amk_tc $8$ (not in LMFDB) 6.2.af_k_ah_ak_bi_ace $8$ (not in LMFDB) 6.2.af_s_abv_dy_aha_ku $8$ (not in LMFDB) 6.2.ad_e_ad_ae_s_abg $8$ (not in LMFDB) 6.2.ad_i_ap_bc_abq_cm $8$ (not in LMFDB) 6.2.ad_m_abb_ci_ady_ge $8$ (not in LMFDB) 6.2.ab_ac_b_c_c_ai $8$ (not in LMFDB) 6.2.ab_e_ab_i_c_q $8$ (not in LMFDB) 6.2.ab_g_ah_s_aw_bo $8$ (not in LMFDB) 6.2.b_ac_ab_c_ac_ai $8$ (not in LMFDB) 6.2.b_e_b_i_ac_q $8$ (not in LMFDB) 6.2.b_g_h_s_w_bo $8$ (not in LMFDB) 6.2.d_e_d_ae_as_abg $8$ (not in LMFDB) 6.2.d_i_p_bc_bq_cm $8$ (not in LMFDB) 6.2.d_m_bb_ci_dy_ge $8$ (not in LMFDB) 6.2.f_k_h_ak_abi_ace $8$ (not in LMFDB) 6.2.f_s_bv_dy_ha_ku $8$ (not in LMFDB) 6.2.h_bc_db_gu_mk_tc $8$ (not in LMFDB) 6.2.af_m_ar_s_au_bc $24$ (not in LMFDB) 6.2.af_q_abp_di_afw_iy $24$ (not in LMFDB) 6.2.af_q_abl_cw_aey_ho $24$ (not in LMFDB) 6.2.ad_g_aj_m_am_q $24$ (not in LMFDB) 6.2.ad_k_av_bs_acu_ei $24$ (not in LMFDB) 6.2.ab_a_ab_g_ae_e $24$ (not in LMFDB) 6.2.ab_e_af_o_aq_bc $24$ (not in LMFDB) 6.2.ab_e_ab_c_i_ai $24$ (not in LMFDB) 6.2.b_a_b_g_e_e $24$ (not in LMFDB) 6.2.b_e_b_c_ai_ai $24$ (not in LMFDB) 6.2.b_e_f_o_q_bc $24$ (not in LMFDB) 6.2.d_g_j_m_m_q $24$ (not in LMFDB) 6.2.d_k_v_bs_cu_ei $24$ (not in LMFDB) 6.2.f_m_r_s_u_bc $24$ (not in LMFDB) 6.2.f_q_bl_cw_ey_ho $24$ (not in LMFDB) 6.2.f_q_bp_di_fw_iy $24$ (not in LMFDB)