# Properties

 Label 5.2.ag_s_abj_ca_acu Base field $\F_{2}$ Dimension $5$ $p$-rank $3$ Ordinary No Supersingular No Simple No Geometrically simple No Primitive Yes Principally polarizable Yes Contains a Jacobian No

## Invariants

 Base field: $\F_{2}$ Dimension: $5$ L-polynomial: $( 1 - x + 2 x^{2} )( 1 - 2 x + 2 x^{2} )^{2}( 1 - x - x^{2} - 2 x^{3} + 4 x^{4} )$ Frobenius angles: $\pm0.0516399385854$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.384973271919$, $\pm0.718306605252$ 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 $A(\F_{q^r})$ 2 1400 37856 2590000 27773482 741977600 32824323394 849649500000 30927679563872 1128089405135000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -3 5 12 33 27 38 123 193 444 1025

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
 The isogeny class factors as 1.2.ac 2 $\times$ 1.2.ab $\times$ 2.2.ab_ab 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}($$$\Q(\sqrt{-1})$$$)$ 1.2.ab : $$\Q(\sqrt{-7})$$. 2.2.ab_ab : $$\Q(\sqrt{-3}, \sqrt{-7})$$.
Endomorphism algebra over $\overline{\F}_{2}$
 The base change of $A$ to $\F_{2^{12}}$ is 1.4096.bv 3 $\times$ 1.4096.ey 2 . The endomorphism algebra for each factor is: 1.4096.bv 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-7})$$$)$ 1.4096.ey 2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$.
All geometric endomorphisms are defined over $\F_{2^{12}}$.
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 $\times$ 1.4.d $\times$ 2.4.ad_f. The endomorphism algebra for each factor is: 1.4.a 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-1})$$$)$ 1.4.d : $$\Q(\sqrt{-7})$$. 2.4.ad_f : $$\Q(\sqrt{-3}, \sqrt{-7})$$.
• Endomorphism algebra over $\F_{2^{3}}$  The base change of $A$ to $\F_{2^{3}}$ is 1.8.af 2 $\times$ 1.8.e 2 $\times$ 1.8.f. The endomorphism algebra for each factor is: 1.8.af 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-7})$$$)$ 1.8.e 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-1})$$$)$ 1.8.f : $$\Q(\sqrt{-7})$$.
• Endomorphism algebra over $\F_{2^{4}}$  The base change of $A$ to $\F_{2^{4}}$ is 1.16.ab $\times$ 1.16.i 2 $\times$ 2.16.b_ap. The endomorphism algebra for each factor is: 1.16.ab : $$\Q(\sqrt{-7})$$. 1.16.i 2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$. 2.16.b_ap : $$\Q(\sqrt{-3}, \sqrt{-7})$$.
• Endomorphism algebra over $\F_{2^{6}}$  The base change of $A$ to $\F_{2^{6}}$ is 1.64.aj 3 $\times$ 1.64.a 2 . The endomorphism algebra for each factor is: 1.64.aj 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-7})$$$)$ 1.64.a 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-1})$$$)$

## 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 5.2.ae_i_an_y_abo $2$ (not in LMFDB) 5.2.ae_i_ad_aq_bo $2$ (not in LMFDB) 5.2.ac_c_ad_i_aq $2$ (not in LMFDB) 5.2.ac_c_d_ae_i $2$ (not in LMFDB) 5.2.a_a_af_e_a $2$ (not in LMFDB) 5.2.a_a_f_e_a $2$ (not in LMFDB) 5.2.c_c_ad_ae_ai $2$ (not in LMFDB) 5.2.c_c_d_i_q $2$ (not in LMFDB) 5.2.e_i_d_aq_abo $2$ (not in LMFDB) 5.2.e_i_n_y_bo $2$ (not in LMFDB) 5.2.g_s_bj_ca_cu $2$ (not in LMFDB) 5.2.ad_j_ar_bi_abw $3$ (not in LMFDB) 5.2.a_a_b_ac_ag $3$ (not in LMFDB) 5.2.d_j_t_bi_cc $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 5.2.ae_i_an_y_abo $2$ (not in LMFDB) 5.2.ae_i_ad_aq_bo $2$ (not in LMFDB) 5.2.ac_c_ad_i_aq $2$ (not in LMFDB) 5.2.ac_c_d_ae_i $2$ (not in LMFDB) 5.2.a_a_af_e_a $2$ (not in LMFDB) 5.2.a_a_f_e_a $2$ (not in LMFDB) 5.2.c_c_ad_ae_ai $2$ (not in LMFDB) 5.2.c_c_d_i_q $2$ (not in LMFDB) 5.2.e_i_d_aq_abo $2$ (not in LMFDB) 5.2.e_i_n_y_bo $2$ (not in LMFDB) 5.2.g_s_bj_ca_cu $2$ (not in LMFDB) 5.2.ad_j_ar_bi_abw $3$ (not in LMFDB) 5.2.a_a_b_ac_ag $3$ (not in LMFDB) 5.2.d_j_t_bi_cc $3$ (not in LMFDB) 5.2.ah_bd_add_go_akm $6$ (not in LMFDB) 5.2.af_r_abp_da_aes $6$ (not in LMFDB) 5.2.af_r_abn_cw_aei $6$ (not in LMFDB) 5.2.ae_i_ap_ba_abm $6$ (not in LMFDB) 5.2.ad_j_at_bi_acc $6$ (not in LMFDB) 5.2.ad_j_an_w_ay $6$ (not in LMFDB) 5.2.ac_c_aj_o_ak $6$ (not in LMFDB) 5.2.ac_c_b_ag_k $6$ (not in LMFDB) 5.2.ab_f_aj_o_aba $6$ (not in LMFDB) 5.2.ab_f_ah_s_aq $6$ (not in LMFDB) 5.2.ab_f_ad_o_ai $6$ (not in LMFDB) 5.2.ab_f_d_c_w $6$ (not in LMFDB) 5.2.a_a_ab_ac_g $6$ (not in LMFDB) 5.2.b_f_ad_c_aw $6$ (not in LMFDB) 5.2.b_f_d_o_i $6$ (not in LMFDB) 5.2.b_f_h_s_q $6$ (not in LMFDB) 5.2.b_f_j_o_ba $6$ (not in LMFDB) 5.2.c_c_ab_ag_ak $6$ (not in LMFDB) 5.2.c_c_j_o_k $6$ (not in LMFDB) 5.2.d_j_n_w_y $6$ (not in LMFDB) 5.2.d_j_r_bi_bw $6$ (not in LMFDB) 5.2.e_i_p_ba_bm $6$ (not in LMFDB) 5.2.f_r_bn_cw_ei $6$ (not in LMFDB) 5.2.f_r_bp_da_es $6$ (not in LMFDB) 5.2.h_bd_dd_go_km $6$ (not in LMFDB) 5.2.ae_k_at_be_abs $8$ (not in LMFDB) 5.2.ac_ac_f_a_ae $8$ (not in LMFDB) 5.2.ac_e_aj_o_au $8$ (not in LMFDB) 5.2.ac_e_b_ag_u $8$ (not in LMFDB) 5.2.ac_g_al_q_abc $8$ (not in LMFDB) 5.2.a_ae_af_e_u $8$ (not in LMFDB) 5.2.a_ae_f_e_au $8$ (not in LMFDB) 5.2.a_c_ad_c_am $8$ (not in LMFDB) 5.2.a_c_d_c_m $8$ (not in LMFDB) 5.2.a_e_af_e_au $8$ (not in LMFDB) 5.2.a_e_f_e_u $8$ (not in LMFDB) 5.2.c_ac_af_a_e $8$ (not in LMFDB) 5.2.c_e_ab_ag_au $8$ (not in LMFDB) 5.2.c_e_j_o_u $8$ (not in LMFDB) 5.2.c_g_l_q_bc $8$ (not in LMFDB) 5.2.e_k_t_be_bs $8$ (not in LMFDB) 5.2.af_l_aj_ak_bg $12$ (not in LMFDB) 5.2.ad_d_ab_ac_g $12$ (not in LMFDB) 5.2.ad_d_b_ac_a $12$ (not in LMFDB) 5.2.ab_ab_ad_c_k $12$ (not in LMFDB) 5.2.ab_ab_d_c_ai $12$ (not in LMFDB) 5.2.b_ab_ad_c_i $12$ (not in LMFDB) 5.2.b_ab_d_c_ak $12$ (not in LMFDB) 5.2.d_d_ab_ac_a $12$ (not in LMFDB) 5.2.d_d_b_ac_ag $12$ (not in LMFDB) 5.2.f_l_j_ak_abg $12$ (not in LMFDB) 5.2.af_t_abv_ds_afs $24$ (not in LMFDB) 5.2.ad_f_ad_ae_m $24$ (not in LMFDB) 5.2.ad_f_ab_ao_bc $24$ (not in LMFDB) 5.2.ad_h_ah_e_c $24$ (not in LMFDB) 5.2.ad_l_av_bs_aci $24$ (not in LMFDB) 5.2.ad_l_at_bo_aby $24$ (not in LMFDB) 5.2.ad_n_az_cg_acy $24$ (not in LMFDB) 5.2.ac_a_b_e_ak $24$ (not in LMFDB) 5.2.ac_e_ah_m_aw $24$ (not in LMFDB) 5.2.ab_af_h_g_au $24$ (not in LMFDB) 5.2.ab_ad_f_e_ao $24$ (not in LMFDB) 5.2.ab_b_ab_a_e $24$ (not in LMFDB) 5.2.ab_b_b_ag_e $24$ (not in LMFDB) 5.2.ab_b_b_a_ac $24$ (not in LMFDB) 5.2.ab_d_ab_ac_e $24$ (not in LMFDB) 5.2.ab_d_ab_e_ac $24$ (not in LMFDB) 5.2.ab_h_ah_y_au $24$ (not in LMFDB) 5.2.ab_h_af_y_ao $24$ (not in LMFDB) 5.2.ab_h_ad_u_ae $24$ (not in LMFDB) 5.2.ab_j_ah_bi_au $24$ (not in LMFDB) 5.2.a_ac_af_e_k $24$ (not in LMFDB) 5.2.a_ac_f_e_ak $24$ (not in LMFDB) 5.2.a_c_af_e_ak $24$ (not in LMFDB) 5.2.a_c_f_e_k $24$ (not in LMFDB) 5.2.b_af_ah_g_u $24$ (not in LMFDB) 5.2.b_ad_af_e_o $24$ (not in LMFDB) 5.2.b_b_ab_ag_ae $24$ (not in LMFDB) 5.2.b_b_ab_a_c $24$ (not in LMFDB) 5.2.b_b_b_a_ae $24$ (not in LMFDB) 5.2.b_d_b_ac_ae $24$ (not in LMFDB) 5.2.b_d_b_e_c $24$ (not in LMFDB) 5.2.b_h_d_u_e $24$ (not in LMFDB) 5.2.b_h_f_y_o $24$ (not in LMFDB) 5.2.b_h_h_y_u $24$ (not in LMFDB) 5.2.b_j_h_bi_u $24$ (not in LMFDB) 5.2.c_a_ab_e_k $24$ (not in LMFDB) 5.2.c_e_h_m_w $24$ (not in LMFDB) 5.2.d_f_b_ao_abc $24$ (not in LMFDB) 5.2.d_f_d_ae_am $24$ (not in LMFDB) 5.2.d_h_h_e_ac $24$ (not in LMFDB) 5.2.d_l_t_bo_by $24$ (not in LMFDB) 5.2.d_l_v_bs_ci $24$ (not in LMFDB) 5.2.d_n_z_cg_cy $24$ (not in LMFDB) 5.2.f_t_bv_ds_fs $24$ (not in LMFDB) 5.2.ai_bh_adn_he_alk $42$ (not in LMFDB) 5.2.ah_w_abn_bs_abw $42$ (not in LMFDB) 5.2.ag_t_abt_di_afe $42$ (not in LMFDB) 5.2.af_k_an_w_abm $42$ (not in LMFDB) 5.2.ae_j_ap_w_abg $42$ (not in LMFDB) 5.2.ad_c_b_i_ay $42$ (not in LMFDB) 5.2.ac_d_ab_ag_k $42$ (not in LMFDB) 5.2.ab_ac_d_c_ag $42$ (not in LMFDB) 5.2.ab_ac_h_e_aq $42$ (not in LMFDB) 5.2.a_b_ad_c_ai $42$ (not in LMFDB) 5.2.a_b_d_c_i $42$ (not in LMFDB) 5.2.b_ac_ah_e_q $42$ (not in LMFDB) 5.2.b_ac_ad_c_g $42$ (not in LMFDB) 5.2.c_d_b_ag_ak $42$ (not in LMFDB) 5.2.d_c_ab_i_y $42$ (not in LMFDB) 5.2.e_j_p_w_bg $42$ (not in LMFDB) 5.2.f_k_n_w_bm $42$ (not in LMFDB) 5.2.g_t_bt_di_fe $42$ (not in LMFDB) 5.2.h_w_bn_bs_bw $42$ (not in LMFDB) 5.2.i_bh_dn_he_lk $42$ (not in LMFDB) 5.2.ag_v_acb_ea_agi $168$ (not in LMFDB) 5.2.af_m_at_ba_abk $168$ (not in LMFDB) 5.2.ae_f_b_ao_bc $168$ (not in LMFDB) 5.2.ae_h_ah_e_ac $168$ (not in LMFDB) 5.2.ae_l_ax_bo_ack $168$ (not in LMFDB) 5.2.ae_n_abf_cg_ado $168$ (not in LMFDB) 5.2.ad_ac_n_a_abc $168$ (not in LMFDB) 5.2.ad_a_h_e_aba $168$ (not in LMFDB) 5.2.ad_e_af_m_aw $168$ (not in LMFDB) 5.2.ad_g_al_q_au $168$ (not in LMFDB) 5.2.ac_f_aj_m_au $168$ (not in LMFDB) 5.2.ab_a_ad_g_ae $168$ (not in LMFDB) 5.2.b_a_d_g_e $168$ (not in LMFDB) 5.2.c_f_j_m_u $168$ (not in LMFDB) 5.2.d_ac_an_a_bc $168$ (not in LMFDB) 5.2.d_a_ah_e_ba $168$ (not in LMFDB) 5.2.d_e_f_m_w $168$ (not in LMFDB) 5.2.d_g_l_q_u $168$ (not in LMFDB) 5.2.e_f_ab_ao_abc $168$ (not in LMFDB) 5.2.e_h_h_e_c $168$ (not in LMFDB) 5.2.e_l_x_bo_ck $168$ (not in LMFDB) 5.2.e_n_bf_cg_do $168$ (not in LMFDB) 5.2.f_m_t_ba_bk $168$ (not in LMFDB) 5.2.g_v_cb_ea_gi $168$ (not in LMFDB)