# Properties

 Label 5.2.ah_w_abn_bs_abw Base Field $\F_{2}$ Dimension $5$ Ordinary No $p$-rank $3$ Principally polarizable Yes Contains a Jacobian No

## Invariants

 Base field: $\F_{2}$ Dimension: $5$ L-polynomial: $( 1 - 2 x + 2 x^{2} )^{2}( 1 - 3 x + 2 x^{2} + x^{3} + 4 x^{4} - 12 x^{5} + 8 x^{6} )$ Frobenius angles: $\pm0.0992589862044$, $\pm0.186455299510$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.757883870938$ Angle rank: $1$ (numerical)

This isogeny class is not simple.

## Newton polygon $p$-rank: $3$ Slopes: $[0, 0, 0, 1/2, 1/2, 1/2, 1/2, 1, 1, 1]$

## 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 $A(\F_{q^r})$ 1 725 50869 5093125 58601341 1585841075 36561324472 832099483125 34659728919361 1116318920211875

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -4 0 11 44 51 87 136 188 506 1015

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
 The isogeny class factors as 1.2.ac 2 $\times$ 3.2.ad_c_b 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}($$$\Q(\sqrt{-1})$$$)$ 3.2.ad_c_b : $$\Q(\zeta_{7})$$.
Endomorphism algebra over $\overline{\F}_{2}$
 The base change of $A$ to $\F_{2^{28}}$ is 1.268435456.blhf 3 $\times$ 1.268435456.bwmi 2 . The endomorphism algebra for each factor is: 1.268435456.blhf 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-7})$$$)$ 1.268435456.bwmi 2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$.
All geometric endomorphisms are defined over $\F_{2^{28}}$.
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 $\times$ 3.4.af_s_abp. The endomorphism algebra for each factor is: 1.4.a 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-1})$$$)$ 3.4.af_s_abp : $$\Q(\zeta_{7})$$.
• Endomorphism algebra over $\F_{2^{4}}$  The base change of $A$ to $\F_{2^{4}}$ is 1.16.i 2 $\times$ 3.16.l_cg_jf. 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 $\infty$. 3.16.l_cg_jf : $$\Q(\zeta_{7})$$.
• Endomorphism algebra over $\F_{2^{7}}$  The base change of $A$ to $\F_{2^{7}}$ is 1.128.aq 2 $\times$ 1.128.n 3 . The endomorphism algebra for each factor is: 1.128.aq 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-1})$$$)$ 1.128.n 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-7})$$$)$
• Endomorphism algebra over $\F_{2^{14}}$  The base change of $A$ to $\F_{2^{14}}$ is 1.16384.a 2 $\times$ 1.16384.dj 3 . The endomorphism algebra for each factor is: 1.16384.a 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-1})$$$)$ 1.16384.dj 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-7})$$$)$

## 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 5.2.ad_c_b_i_ay $2$ (not in LMFDB) 5.2.ab_ac_h_e_aq $2$ (not in LMFDB) 5.2.b_ac_ah_e_q $2$ (not in LMFDB) 5.2.d_c_ab_i_y $2$ (not in LMFDB) 5.2.h_w_bn_bs_bw $2$ (not in LMFDB) 5.2.ab_ac_d_c_ag $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 5.2.ad_c_b_i_ay $2$ (not in LMFDB) 5.2.ab_ac_h_e_aq $2$ (not in LMFDB) 5.2.b_ac_ah_e_q $2$ (not in LMFDB) 5.2.d_c_ab_i_y $2$ (not in LMFDB) 5.2.h_w_bn_bs_bw $2$ (not in LMFDB) 5.2.ab_ac_d_c_ag $3$ (not in LMFDB) 5.2.af_k_an_w_abm $6$ (not in LMFDB) 5.2.b_ac_ad_c_g $6$ (not in LMFDB) 5.2.f_k_n_w_bm $6$ (not in LMFDB) 5.2.ah_bd_add_go_akm $7$ (not in LMFDB) 5.2.a_b_d_c_i $7$ (not in LMFDB) 5.2.af_m_at_ba_abk $8$ (not in LMFDB) 5.2.ad_ac_n_a_abc $8$ (not in LMFDB) 5.2.ad_g_al_q_au $8$ (not in LMFDB) 5.2.ab_a_ad_g_ae $8$ (not in LMFDB) 5.2.b_a_d_g_e $8$ (not in LMFDB) 5.2.d_ac_an_a_bc $8$ (not in LMFDB) 5.2.d_g_l_q_u $8$ (not in LMFDB) 5.2.f_m_t_ba_bk $8$ (not in LMFDB) 5.2.ai_bh_adn_he_alk $14$ (not in LMFDB) 5.2.af_r_abn_cw_aei $14$ (not in LMFDB) 5.2.ae_j_ap_w_abg $14$ (not in LMFDB) 5.2.ad_j_ar_bi_abw $14$ (not in LMFDB) 5.2.ad_j_an_w_ay $14$ (not in LMFDB) 5.2.ab_f_ah_s_aq $14$ (not in LMFDB) 5.2.ab_f_ad_o_ai $14$ (not in LMFDB) 5.2.a_b_ad_c_ai $14$ (not in LMFDB) 5.2.b_f_d_o_i $14$ (not in LMFDB) 5.2.b_f_h_s_q $14$ (not in LMFDB) 5.2.d_j_n_w_y $14$ (not in LMFDB) 5.2.d_j_r_bi_bw $14$ (not in LMFDB) 5.2.e_j_p_w_bg $14$ (not in LMFDB) 5.2.f_r_bn_cw_ei $14$ (not in LMFDB) 5.2.h_bd_dd_go_km $14$ (not in LMFDB) 5.2.i_bh_dn_he_lk $14$ (not in LMFDB) 5.2.ae_i_ad_aq_bo $21$ (not in LMFDB) 5.2.ab_f_d_c_w $21$ (not in LMFDB) 5.2.c_c_j_o_k $21$ (not in LMFDB) 5.2.g_t_bt_di_fe $21$ (not in LMFDB) 5.2.ad_a_h_e_aba $24$ (not in LMFDB) 5.2.ad_e_af_m_aw $24$ (not in LMFDB) 5.2.d_a_ah_e_ba $24$ (not in LMFDB) 5.2.d_e_f_m_w $24$ (not in LMFDB) 5.2.af_l_aj_ak_bg $28$ (not in LMFDB) 5.2.ad_d_b_ac_a $28$ (not in LMFDB) 5.2.ab_ab_d_c_ai $28$ (not in LMFDB) 5.2.b_ab_ad_c_i $28$ (not in LMFDB) 5.2.d_d_ab_ac_a $28$ (not in LMFDB) 5.2.f_l_j_ak_abg $28$ (not in LMFDB) 5.2.ag_s_abj_ca_acu $42$ (not in LMFDB) 5.2.ag_t_abt_di_afe $42$ (not in LMFDB) 5.2.af_r_abp_da_aes $42$ (not in LMFDB) 5.2.ae_i_ap_ba_abm $42$ (not in LMFDB) 5.2.ae_i_an_y_abo $42$ (not in LMFDB) 5.2.ad_j_at_bi_acc $42$ (not in LMFDB) 5.2.ac_c_aj_o_ak $42$ (not in LMFDB) 5.2.ac_c_ad_i_aq $42$ (not in LMFDB) 5.2.ac_c_b_ag_k $42$ (not in LMFDB) 5.2.ac_c_d_ae_i $42$ (not in LMFDB) 5.2.ac_d_ab_ag_k $42$ (not in LMFDB) 5.2.ab_f_aj_o_aba $42$ (not in LMFDB) 5.2.a_a_af_e_a $42$ (not in LMFDB) 5.2.a_a_ab_ac_g $42$ (not in LMFDB) 5.2.a_a_b_ac_ag $42$ (not in LMFDB) 5.2.a_a_f_e_a $42$ (not in LMFDB) 5.2.b_f_ad_c_aw $42$ (not in LMFDB) 5.2.b_f_j_o_ba $42$ (not in LMFDB) 5.2.c_c_ad_ae_ai $42$ (not in LMFDB) 5.2.c_c_ab_ag_ak $42$ (not in LMFDB) 5.2.c_c_d_i_q $42$ (not in LMFDB) 5.2.c_d_b_ag_ak $42$ (not in LMFDB) 5.2.d_j_t_bi_cc $42$ (not in LMFDB) 5.2.e_i_d_aq_abo $42$ (not in LMFDB) 5.2.e_i_n_y_bo $42$ (not in LMFDB) 5.2.e_i_p_ba_bm $42$ (not in LMFDB) 5.2.f_r_bp_da_es $42$ (not in LMFDB) 5.2.g_s_bj_ca_cu $42$ (not in LMFDB) 5.2.ag_v_acb_ea_agi $56$ (not in LMFDB) 5.2.af_t_abv_ds_afs $56$ (not in LMFDB) 5.2.ae_f_b_ao_bc $56$ (not in LMFDB) 5.2.ae_n_abf_cg_ado $56$ (not in LMFDB) 5.2.ad_f_ad_ae_m $56$ (not in LMFDB) 5.2.ad_f_ab_ao_bc $56$ (not in LMFDB) 5.2.ad_l_av_bs_aci $56$ (not in LMFDB) 5.2.ad_n_az_cg_acy $56$ (not in LMFDB) 5.2.ac_f_aj_m_au $56$ (not in LMFDB) 5.2.ab_af_h_g_au $56$ (not in LMFDB) 5.2.ab_b_ab_a_e $56$ (not in LMFDB) 5.2.ab_b_b_ag_e $56$ (not in LMFDB) 5.2.ab_d_ab_ac_e $56$ (not in LMFDB) 5.2.ab_h_ah_y_au $56$ (not in LMFDB) 5.2.ab_h_ad_u_ae $56$ (not in LMFDB) 5.2.ab_j_ah_bi_au $56$ (not in LMFDB) 5.2.b_af_ah_g_u $56$ (not in LMFDB) 5.2.b_b_ab_ag_ae $56$ (not in LMFDB) 5.2.b_b_b_a_ae $56$ (not in LMFDB) 5.2.b_d_b_ac_ae $56$ (not in LMFDB) 5.2.b_h_d_u_e $56$ (not in LMFDB) 5.2.b_h_h_y_u $56$ (not in LMFDB) 5.2.b_j_h_bi_u $56$ (not in LMFDB) 5.2.c_f_j_m_u $56$ (not in LMFDB) 5.2.d_f_b_ao_abc $56$ (not in LMFDB) 5.2.d_f_d_ae_am $56$ (not in LMFDB) 5.2.d_l_v_bs_ci $56$ (not in LMFDB) 5.2.d_n_z_cg_cy $56$ (not in LMFDB) 5.2.e_f_ab_ao_abc $56$ (not in LMFDB) 5.2.e_n_bf_cg_do $56$ (not in LMFDB) 5.2.f_t_bv_ds_fs $56$ (not in LMFDB) 5.2.g_v_cb_ea_gi $56$ (not in LMFDB) 5.2.ad_d_ab_ac_g $84$ (not in LMFDB) 5.2.ab_ab_ad_c_k $84$ (not in LMFDB) 5.2.b_ab_d_c_ak $84$ (not in LMFDB) 5.2.d_d_b_ac_ag $84$ (not in LMFDB) 5.2.ae_h_ah_e_ac $168$ (not in LMFDB) 5.2.ae_k_at_be_abs $168$ (not in LMFDB) 5.2.ae_l_ax_bo_ack $168$ (not in LMFDB) 5.2.ad_h_ah_e_c $168$ (not in LMFDB) 5.2.ad_l_at_bo_aby $168$ (not in LMFDB) 5.2.ac_ac_f_a_ae $168$ (not in LMFDB) 5.2.ac_a_b_e_ak $168$ (not in LMFDB) 5.2.ac_e_aj_o_au $168$ (not in LMFDB) 5.2.ac_e_ah_m_aw $168$ (not in LMFDB) 5.2.ac_e_b_ag_u $168$ (not in LMFDB) 5.2.ac_g_al_q_abc $168$ (not in LMFDB) 5.2.ab_ad_f_e_ao $168$ (not in LMFDB) 5.2.ab_b_b_a_ac $168$ (not in LMFDB) 5.2.ab_d_ab_e_ac $168$ (not in LMFDB) 5.2.ab_h_af_y_ao $168$ (not in LMFDB) 5.2.a_ae_af_e_u $168$ (not in LMFDB) 5.2.a_ae_f_e_au $168$ (not in LMFDB) 5.2.a_ac_af_e_k $168$ (not in LMFDB) 5.2.a_ac_f_e_ak $168$ (not in LMFDB) 5.2.a_c_af_e_ak $168$ (not in LMFDB) 5.2.a_c_ad_c_am $168$ (not in LMFDB) 5.2.a_c_d_c_m $168$ (not in LMFDB) 5.2.a_c_f_e_k $168$ (not in LMFDB) 5.2.a_e_af_e_au $168$ (not in LMFDB) 5.2.a_e_f_e_u $168$ (not in LMFDB) 5.2.b_ad_af_e_o $168$ (not in LMFDB) 5.2.b_b_ab_a_c $168$ (not in LMFDB) 5.2.b_d_b_e_c $168$ (not in LMFDB) 5.2.b_h_f_y_o $168$ (not in LMFDB) 5.2.c_ac_af_a_e $168$ (not in LMFDB) 5.2.c_a_ab_e_k $168$ (not in LMFDB) 5.2.c_e_ab_ag_au $168$ (not in LMFDB) 5.2.c_e_h_m_w $168$ (not in LMFDB) 5.2.c_e_j_o_u $168$ (not in LMFDB) 5.2.c_g_l_q_bc $168$ (not in LMFDB) 5.2.d_h_h_e_ac $168$ (not in LMFDB) 5.2.d_l_t_bo_by $168$ (not in LMFDB) 5.2.e_h_h_e_c $168$ (not in LMFDB) 5.2.e_k_t_be_bs $168$ (not in LMFDB) 5.2.e_l_x_bo_ck $168$ (not in LMFDB)