# Properties

 Label 6.2.aj_bo_aem_jp_aqq_yy Base Field $\F_{2}$ Dimension $6$ Ordinary No $p$-rank $4$ Principally polarizable Yes Contains a Jacobian No

## Invariants

 Base field: $\F_{2}$ Dimension: $6$ L-polynomial: $( 1 - 2 x + 2 x^{2} )^{2}( 1 - 5 x + 12 x^{2} - 20 x^{3} + 29 x^{4} - 40 x^{5} + 48 x^{6} - 40 x^{7} + 16 x^{8} )$ Frobenius angles: $\pm0.0635622003031$, $\pm0.165221137389$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.365221137389$, $\pm0.663562200303$ Angle rank: $2$ (numerical)

This isogeny class is not simple.

## Newton polygon $p$-rank: $4$ Slopes: $[0, 0, 0, 0, 1/2, 1/2, 1/2, 1/2, 1, 1, 1, 1]$

## 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 $A(\F_{q^r})$ 1 5275 314509 55519375 1759622051 51430084225 4397110890089 283427797359375 15777212909914651 1151200833717075625

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -6 4 12 36 49 46 127 260 444 1019

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
 The isogeny class factors as 1.2.ac 2 $\times$ 4.2.af_m_au_bd 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 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-1})$$$)$ 4.2.af_m_au_bd : 8.0.13140625.1.
Endomorphism algebra over $\overline{\F}_{2}$
 The base change of $A$ to $\F_{2^{20}}$ is 1.1048576.dau 2 $\times$ 2.1048576.dth_ibxft 2 . The endomorphism algebra for each factor is: 1.1048576.dau 2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$. 2.1048576.dth_ibxft 2 : $\mathrm{M}_{2}($4.0.3625.1$)$
All geometric endomorphisms are defined over $\F_{2^{20}}$.
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 2 $\times$ 4.4.ab_c_ai_z. The endomorphism algebra for each factor is: 1.4.a 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-1})$$$)$ 4.4.ab_c_ai_z : 8.0.13140625.1.
• Endomorphism algebra over $\F_{2^{4}}$  The base change of $A$ to $\F_{2^{4}}$ is 1.16.i 2 $\times$ 4.16.d_bm_bk_yb. The endomorphism algebra for each factor is: 1.16.i 2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$. 4.16.d_bm_bk_yb : 8.0.13140625.1.
• Endomorphism algebra over $\F_{2^{5}}$  The base change of $A$ to $\F_{2^{5}}$ is 1.32.i 2 $\times$ 4.32.a_ad_a_bwv. The endomorphism algebra for each factor is: 1.32.i 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-1})$$$)$ 4.32.a_ad_a_bwv : 8.0.13140625.1.
• Endomorphism algebra over $\F_{2^{10}}$  The base change of $A$ to $\F_{2^{10}}$ is 1.1024.a 2 $\times$ 2.1024.ad_bwv 2 . The endomorphism algebra for each factor is: 1.1024.a 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-1})$$$)$ 2.1024.ad_bwv 2 : $\mathrm{M}_{2}($4.0.3625.1$)$

## 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_m_au_bh_aci_ds $2$ (not in LMFDB) 6.2.ab_a_ae_j_ai_i $2$ (not in LMFDB) 6.2.b_a_e_j_i_i $2$ (not in LMFDB) 6.2.f_m_u_bh_ci_ds $2$ (not in LMFDB) 6.2.j_bo_em_jp_qq_yy $2$ (not in LMFDB) 6.2.ad_e_ac_ad_g_ag $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 6.2.af_m_au_bh_aci_ds $2$ (not in LMFDB) 6.2.ab_a_ae_j_ai_i $2$ (not in LMFDB) 6.2.b_a_e_j_i_i $2$ (not in LMFDB) 6.2.f_m_u_bh_ci_ds $2$ (not in LMFDB) 6.2.j_bo_em_jp_qq_yy $2$ (not in LMFDB) 6.2.ad_e_ac_ad_g_ag $3$ (not in LMFDB) 6.2.ae_f_e_al_am_bw $5$ (not in LMFDB) 6.2.ae_k_av_bn_ack_dk $5$ (not in LMFDB) 6.2.ae_k_al_ab_bm_acu $5$ (not in LMFDB) 6.2.b_a_e_j_i_i $5$ (not in LMFDB) 6.2.ah_y_acg_en_ahy_mc $6$ (not in LMFDB) 6.2.d_e_c_ad_ag_ag $6$ (not in LMFDB) 6.2.h_y_cg_en_hy_mc $6$ (not in LMFDB) 6.2.ah_ba_acq_fl_ajm_oi $8$ (not in LMFDB) 6.2.af_i_a_ap_u_au $8$ (not in LMFDB) 6.2.af_q_abo_dd_afk_ie $8$ (not in LMFDB) 6.2.ad_g_am_v_abi_ca $8$ (not in LMFDB) 6.2.d_g_m_v_bi_ca $8$ (not in LMFDB) 6.2.f_i_a_ap_au_au $8$ (not in LMFDB) 6.2.f_q_bo_dd_fk_ie $8$ (not in LMFDB) 6.2.h_ba_cq_fl_jm_oi $8$ (not in LMFDB) 6.2.a_ad_a_n_a_ay $10$ (not in LMFDB) 6.2.a_c_af_d_ak_q $10$ (not in LMFDB) 6.2.a_c_f_d_k_q $10$ (not in LMFDB) 6.2.e_f_ae_al_m_bw $10$ (not in LMFDB) 6.2.e_k_l_ab_abm_acu $10$ (not in LMFDB) 6.2.e_k_v_bn_ck_dk $10$ (not in LMFDB) 6.2.c_ab_ac_h_g_ag $15$ (not in LMFDB) 6.2.c_e_d_ad_ao_aba $15$ (not in LMFDB) 6.2.c_e_n_r_ba_cc $15$ (not in LMFDB) 6.2.h_y_cg_en_hy_mc $15$ (not in LMFDB) 6.2.ae_l_au_bl_aci_ds $20$ (not in LMFDB) 6.2.a_d_a_n_a_y $20$ (not in LMFDB) 6.2.e_l_u_bl_ci_ds $20$ (not in LMFDB) 6.2.af_k_ak_j_au_bm $24$ (not in LMFDB) 6.2.af_o_abe_cf_adw_fy $24$ (not in LMFDB) 6.2.f_k_k_j_u_bm $24$ (not in LMFDB) 6.2.f_o_be_cf_dw_fy $24$ (not in LMFDB) 6.2.ac_ab_c_h_ag_ag $30$ (not in LMFDB) 6.2.ac_e_an_r_aba_cc $30$ (not in LMFDB) 6.2.ac_e_ad_ad_o_aba $30$ (not in LMFDB) 6.2.ac_b_c_b_ag_m $40$ (not in LMFDB) 6.2.ac_g_an_v_abk_ca $40$ (not in LMFDB) 6.2.ac_g_ad_b_y_abc $40$ (not in LMFDB) 6.2.ac_h_ak_z_abe_ci $40$ (not in LMFDB) 6.2.a_ah_a_z_a_aci $40$ (not in LMFDB) 6.2.a_ac_af_af_k_u $40$ (not in LMFDB) 6.2.a_ac_f_af_ak_u $40$ (not in LMFDB) 6.2.a_ab_a_b_a_am $40$ (not in LMFDB) 6.2.a_b_a_b_a_m $40$ (not in LMFDB) 6.2.a_g_af_l_abe_m $40$ (not in LMFDB) 6.2.a_g_f_l_be_m $40$ (not in LMFDB) 6.2.a_h_a_z_a_ci $40$ (not in LMFDB) 6.2.c_b_ac_b_g_m $40$ (not in LMFDB) 6.2.c_g_d_b_ay_abc $40$ (not in LMFDB) 6.2.c_g_n_v_bk_ca $40$ (not in LMFDB) 6.2.c_h_k_z_be_ci $40$ (not in LMFDB) 6.2.ac_f_ak_t_abe_bq $60$ (not in LMFDB) 6.2.c_f_k_t_be_bq $60$ (not in LMFDB) 6.2.a_af_a_t_a_abq $120$ (not in LMFDB) 6.2.a_ab_a_h_a_ag $120$ (not in LMFDB) 6.2.a_a_af_ab_a_s $120$ (not in LMFDB) 6.2.a_a_f_ab_a_s $120$ (not in LMFDB) 6.2.a_b_a_h_a_g $120$ (not in LMFDB) 6.2.a_e_af_h_au_o $120$ (not in LMFDB) 6.2.a_e_f_h_u_o $120$ (not in LMFDB) 6.2.a_f_a_t_a_bq $120$ (not in LMFDB)