# Properties

 Label 5.2.ag_r_abd_bj_abq Base Field $\F_{2}$ Dimension $5$ Ordinary No $p$-rank $4$ Principally polarizable Yes Contains a Jacobian No

## Invariants

 Base field: $\F_{2}$ Dimension: $5$ L-polynomial: $( 1 - 2 x + 2 x^{2} )( 1 - x + 2 x^{2} )( 1 - 3 x + 2 x^{2} + x^{3} + 4 x^{4} - 12 x^{5} + 8 x^{6} )$ Frobenius angles: $\pm0.0992589862044$, $\pm0.186455299510$, $\pm0.250000000000$, $\pm0.384973271919$, $\pm0.757883870938$ Angle rank: $1$ (numerical)

This isogeny class is not simple.

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

## 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 $A(\F_{q^r})$ 2 1160 54782 3259600 31444622 1366263080 45944319248 1065087338400 37325861913158 1054240697331800

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -3 3 12 35 32 78 165 251 543 958

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
 The isogeny class factors as 1.2.ac $\times$ 1.2.ab $\times$ 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:
Endomorphism algebra over $\overline{\F}_{2}$
 The base change of $A$ to $\F_{2^{28}}$ is 1.268435456.blhf 4 $\times$ 1.268435456.bwmi. The endomorphism algebra for each factor is: 1.268435456.blhf 4 : $\mathrm{M}_{4}($$$\Q(\sqrt{-7})$$$)$ 1.268435456.bwmi : 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 $\times$ 1.4.d $\times$ 3.4.af_s_abp. The endomorphism algebra for each factor is:
• Endomorphism algebra over $\F_{2^{4}}$  The base change of $A$ to $\F_{2^{4}}$ is 1.16.ab $\times$ 1.16.i $\times$ 3.16.l_cg_jf. The endomorphism algebra for each factor is: 1.16.ab : $$\Q(\sqrt{-7})$$. 1.16.i : 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 $\times$ 1.128.n 4 . The endomorphism algebra for each factor is: 1.128.aq : $$\Q(\sqrt{-1})$$. 1.128.n 4 : $\mathrm{M}_{4}($$$\Q(\sqrt{-7})$$$)$
• Endomorphism algebra over $\F_{2^{14}}$  The base change of $A$ to $\F_{2^{14}}$ is 1.16384.a $\times$ 1.16384.dj 4 . The endomorphism algebra for each factor is: 1.16384.a : $$\Q(\sqrt{-1})$$. 1.16384.dj 4 : $\mathrm{M}_{4}($$$\Q(\sqrt{-7})$$$)$

## 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.ae_h_aj_r_abe $2$ (not in LMFDB) 5.2.ac_b_ab_h_ao $2$ (not in LMFDB) 5.2.a_ab_af_f_g $2$ (not in LMFDB) 5.2.a_ab_f_f_ag $2$ (not in LMFDB) 5.2.c_b_b_h_o $2$ (not in LMFDB) 5.2.e_h_j_r_be $2$ (not in LMFDB) 5.2.g_r_bd_bj_bq $2$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 5.2.ae_h_aj_r_abe $2$ (not in LMFDB) 5.2.ac_b_ab_h_ao $2$ (not in LMFDB) 5.2.a_ab_af_f_g $2$ (not in LMFDB) 5.2.a_ab_f_f_ag $2$ (not in LMFDB) 5.2.c_b_b_h_o $2$ (not in LMFDB) 5.2.e_h_j_r_be $2$ (not in LMFDB) 5.2.g_r_bd_bj_bq $2$ (not in LMFDB) 5.2.ag_y_acm_fd_aic $7$ (not in LMFDB) 5.2.b_d_g_h_o $7$ (not in LMFDB) 5.2.ae_j_ap_v_abc $8$ (not in LMFDB) 5.2.ac_d_ah_l_am $8$ (not in LMFDB) 5.2.c_d_h_l_m $8$ (not in LMFDB) 5.2.e_j_p_v_bc $8$ (not in LMFDB) 5.2.ah_bb_acu_fp_aiw $14$ (not in LMFDB) 5.2.af_p_abi_cl_adu $14$ (not in LMFDB) 5.2.ae_o_abe_ch_adi $14$ (not in LMFDB) 5.2.ad_h_am_r_aba $14$ (not in LMFDB) 5.2.ac_i_am_bd_abi $14$ (not in LMFDB) 5.2.ac_i_ai_v_ao $14$ (not in LMFDB) 5.2.ab_d_ag_h_ao $14$ (not in LMFDB) 5.2.a_g_ac_t_ag $14$ (not in LMFDB) 5.2.a_g_c_t_g $14$ (not in LMFDB) 5.2.c_i_i_v_o $14$ (not in LMFDB) 5.2.c_i_m_bd_bi $14$ (not in LMFDB) 5.2.d_h_m_r_ba $14$ (not in LMFDB) 5.2.e_o_be_ch_di $14$ (not in LMFDB) 5.2.f_p_bi_cl_du $14$ (not in LMFDB) 5.2.g_y_cm_fd_ic $14$ (not in LMFDB) 5.2.h_bb_cu_fp_iw $14$ (not in LMFDB) 5.2.ad_g_ab_al_be $21$ (not in LMFDB) 5.2.a_ad_i_h_as $21$ (not in LMFDB) 5.2.ae_i_ag_ah_w $28$ (not in LMFDB) 5.2.ac_ae_m_f_abi $28$ (not in LMFDB) 5.2.ac_c_a_ab_c $28$ (not in LMFDB) 5.2.a_a_ac_b_g $28$ (not in LMFDB) 5.2.a_a_c_b_ag $28$ (not in LMFDB) 5.2.c_ae_am_f_bi $28$ (not in LMFDB) 5.2.c_c_a_ab_ac $28$ (not in LMFDB) 5.2.e_i_g_ah_aw $28$ (not in LMFDB) 5.2.ab_ab_b_d_ak $35$ (not in LMFDB) 5.2.af_o_abb_bp_acg $42$ (not in LMFDB) 5.2.ae_f_ae_p_abi $42$ (not in LMFDB) 5.2.ad_g_al_t_abe $42$ (not in LMFDB) 5.2.ac_ab_g_ab_ak $42$ (not in LMFDB) 5.2.ab_c_ah_j_ak $42$ (not in LMFDB) 5.2.ab_c_ad_f_ao $42$ (not in LMFDB) 5.2.ab_c_d_ab_k $42$ (not in LMFDB) 5.2.a_ad_ai_h_s $42$ (not in LMFDB) 5.2.b_c_ad_ab_ak $42$ (not in LMFDB) 5.2.b_c_d_f_o $42$ (not in LMFDB) 5.2.b_c_h_j_k $42$ (not in LMFDB) 5.2.c_ab_ag_ab_k $42$ (not in LMFDB) 5.2.d_g_b_al_abe $42$ (not in LMFDB) 5.2.d_g_l_t_be $42$ (not in LMFDB) 5.2.e_f_e_p_bi $42$ (not in LMFDB) 5.2.f_o_bb_bp_cg $42$ (not in LMFDB) 5.2.af_r_abq_dd_aey $56$ (not in LMFDB) 5.2.ae_q_abk_cz_aei $56$ (not in LMFDB) 5.2.ad_j_au_bj_ace $56$ (not in LMFDB) 5.2.ac_c_a_b_ac $56$ (not in LMFDB) 5.2.ac_e_ac_ad_i $56$ (not in LMFDB) 5.2.ac_k_ao_bn_abo $56$ (not in LMFDB) 5.2.a_ae_a_f_a $56$ (not in LMFDB) 5.2.a_c_a_ab_a $56$ (not in LMFDB) 5.2.a_c_a_b_a $56$ (not in LMFDB) 5.2.a_i_a_bd_a $56$ (not in LMFDB) 5.2.c_c_a_b_c $56$ (not in LMFDB) 5.2.c_e_c_ad_ai $56$ (not in LMFDB) 5.2.c_k_o_bn_bo $56$ (not in LMFDB) 5.2.d_j_u_bj_ce $56$ (not in LMFDB) 5.2.e_q_bk_cz_ei $56$ (not in LMFDB) 5.2.f_r_bq_dd_ey $56$ (not in LMFDB) 5.2.ad_d_d_aj_o $70$ (not in LMFDB) 5.2.b_ab_ab_d_k $70$ (not in LMFDB) 5.2.d_d_ad_aj_ao $70$ (not in LMFDB) 5.2.ad_a_h_b_as $84$ (not in LMFDB) 5.2.ac_f_ag_l_ak $84$ (not in LMFDB) 5.2.ab_ae_j_f_aba $84$ (not in LMFDB) 5.2.b_ae_aj_f_ba $84$ (not in LMFDB) 5.2.c_f_g_l_k $84$ (not in LMFDB) 5.2.d_a_ah_b_s $84$ (not in LMFDB) 5.2.ad_i_ap_x_abk $168$ (not in LMFDB) 5.2.ac_b_ag_l_ai $168$ (not in LMFDB) 5.2.ab_ac_ab_d_e $168$ (not in LMFDB) 5.2.ab_e_ah_j_au $168$ (not in LMFDB) 5.2.ab_e_d_ab_u $168$ (not in LMFDB) 5.2.a_ab_a_ab_a $168$ (not in LMFDB) 5.2.a_f_a_l_a $168$ (not in LMFDB) 5.2.b_ac_b_d_ae $168$ (not in LMFDB) 5.2.b_e_ad_ab_au $168$ (not in LMFDB) 5.2.b_e_h_j_u $168$ (not in LMFDB) 5.2.c_b_g_l_i $168$ (not in LMFDB) 5.2.d_i_p_x_bk $168$ (not in LMFDB) 5.2.ab_b_b_ad_m $280$ (not in LMFDB) 5.2.b_b_ab_ad_am $280$ (not in LMFDB)