# Properties

 Label 5.3.ak_by_agm_qh_abfw Base Field $\F_{3}$ Dimension $5$ Ordinary No $p$-rank $2$ Principally polarizable Yes Contains a Jacobian No

## Invariants

 Base field: $\F_{3}$ Dimension: $5$ L-polynomial: $( 1 + 3 x^{2} )( 1 - 3 x + 3 x^{2} )^{2}( 1 - 4 x + 8 x^{2} - 12 x^{3} + 9 x^{4} )$ Frobenius angles: $\pm0.0540867239847$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.445913276015$, $\pm0.5$ Angle rank: $1$ (numerical)

This isogeny class is not simple.

## Newton polygon

 $p$-rank: $2$ Slopes: $[0, 0, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1, 1]$

## 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 $A(\F_{q^r})$ 8 53312 13741952 2450646016 883113924328 256413830758400 55175182657623064 12035628397913702400 2911251624101322728576 717921606084570114339392

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -6 10 24 54 254 892 2402 6494 19392 59050

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ad 2 $\times$ 1.3.a $\times$ 2.3.ae_i and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: 1.3.ad 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$ 1.3.a : $$\Q(\sqrt{-3})$$. 2.3.ae_i : $$\Q(\zeta_{8})$$.
Endomorphism algebra over $\overline{\F}_{3}$
 The base change of $A$ to $\F_{3^{12}}$ is 1.531441.acec 3 $\times$ 1.531441.zi 2 . The endomorphism algebra for each factor is: 1.531441.acec 3 : $\mathrm{M}_{3}(B)$, where $B$ is the quaternion algebra over $$\Q$$ ramified at $3$ and $\infty$. 1.531441.zi 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-2})$$$)$
All geometric endomorphisms are defined over $\F_{3^{12}}$.
Remainder of endomorphism lattice by field
• Endomorphism algebra over $\F_{3^{2}}$  The base change of $A$ to $\F_{3^{2}}$ is 1.9.ad 2 $\times$ 1.9.g $\times$ 2.9.a_ao. The endomorphism algebra for each factor is: 1.9.ad 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$ 1.9.g : the quaternion algebra over $$\Q$$ ramified at $3$ and $\infty$. 2.9.a_ao : $$\Q(\zeta_{8})$$.
• Endomorphism algebra over $\F_{3^{3}}$  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a 3 $\times$ 2.27.ae_i. The endomorphism algebra for each factor is: 1.27.a 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-3})$$$)$ 2.27.ae_i : $$\Q(\zeta_{8})$$.
• Endomorphism algebra over $\F_{3^{4}}$  The base change of $A$ to $\F_{3^{4}}$ is 1.81.as $\times$ 1.81.ao 2 $\times$ 1.81.j 2 . The endomorphism algebra for each factor is: 1.81.as : the quaternion algebra over $$\Q$$ ramified at $3$ and $\infty$. 1.81.ao 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-2})$$$)$ 1.81.j 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$
• Endomorphism algebra over $\F_{3^{6}}$  The base change of $A$ to $\F_{3^{6}}$ is 1.729.cc 3 $\times$ 2.729.a_zi. The endomorphism algebra for each factor is: 1.729.cc 3 : $\mathrm{M}_{3}(B)$, where $B$ is the quaternion algebra over $$\Q$$ ramified at $3$ and $\infty$. 2.729.a_zi : $$\Q(\zeta_{8})$$.

## 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 5.3.ae_i_am_j_a $2$ (not in LMFDB) 5.3.ac_c_a_aj_bk $2$ (not in LMFDB) 5.3.c_c_a_aj_abk $2$ (not in LMFDB) 5.3.e_i_m_j_a $2$ (not in LMFDB) 5.3.k_by_gm_qh_bfw $2$ (not in LMFDB) 5.3.an_dc_alx_bgf_acmk $3$ (not in LMFDB) 5.3.ah_u_abb_j_s $3$ (not in LMFDB) 5.3.ah_bd_adm_ii_apy $3$ (not in LMFDB) 5.3.ae_r_abw_ee_aii $3$ (not in LMFDB) 5.3.ab_ae_d_j_as $3$ (not in LMFDB) 5.3.ab_f_ag_a_as $3$ (not in LMFDB) 5.3.f_i_ad_abb_acc $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 5.3.ae_i_am_j_a $2$ (not in LMFDB) 5.3.ac_c_a_aj_bk $2$ (not in LMFDB) 5.3.c_c_a_aj_abk $2$ (not in LMFDB) 5.3.e_i_m_j_a $2$ (not in LMFDB) 5.3.k_by_gm_qh_bfw $2$ (not in LMFDB) 5.3.an_dc_alx_bgf_acmk $3$ (not in LMFDB) 5.3.ah_u_abb_j_s $3$ (not in LMFDB) 5.3.ah_bd_adm_ii_apy $3$ (not in LMFDB) 5.3.ae_r_abw_ee_aii $3$ (not in LMFDB) 5.3.ab_ae_d_j_as $3$ (not in LMFDB) 5.3.ab_f_ag_a_as $3$ (not in LMFDB) 5.3.f_i_ad_abb_acc $3$ (not in LMFDB) 5.3.ae_o_abk_cx_afo $4$ (not in LMFDB) 5.3.e_o_bk_cx_fo $4$ (not in LMFDB) 5.3.af_i_d_abb_cc $6$ (not in LMFDB) 5.3.b_ae_ad_j_s $6$ (not in LMFDB) 5.3.b_f_g_a_s $6$ (not in LMFDB) 5.3.e_r_bw_ee_ii $6$ (not in LMFDB) 5.3.h_u_bb_j_as $6$ (not in LMFDB) 5.3.h_bd_dm_ii_py $6$ (not in LMFDB) 5.3.n_dc_lx_bgf_cmk $6$ (not in LMFDB) 5.3.ak_ca_agy_rr_abiq $8$ (not in LMFDB) 5.3.ag_q_ay_bb_abk $8$ (not in LMFDB) 5.3.ag_u_abw_dv_agy $8$ (not in LMFDB) 5.3.ae_k_am_j_a $8$ (not in LMFDB) 5.3.ae_q_abk_dj_afo $8$ (not in LMFDB) 5.3.ac_e_am_bb_abk $8$ (not in LMFDB) 5.3.a_ac_a_j_a $8$ (not in LMFDB) 5.3.a_c_a_j_a $8$ (not in LMFDB) 5.3.a_e_a_p_a $8$ (not in LMFDB) 5.3.a_i_a_bn_a $8$ (not in LMFDB) 5.3.c_e_m_bb_bk $8$ (not in LMFDB) 5.3.e_k_m_j_a $8$ (not in LMFDB) 5.3.e_q_bk_dj_fo $8$ (not in LMFDB) 5.3.g_q_y_bb_bk $8$ (not in LMFDB) 5.3.g_u_bw_dv_gy $8$ (not in LMFDB) 5.3.k_ca_gy_rr_biq $8$ (not in LMFDB) 5.3.ae_i_av_bt_acu $9$ (not in LMFDB) 5.3.ae_i_ad_abb_cu $9$ (not in LMFDB) 5.3.ah_r_ag_aci_gg $12$ (not in LMFDB) 5.3.ah_ba_acr_fr_akk $12$ (not in LMFDB) 5.3.ae_f_a_ay_cu $12$ (not in LMFDB) 5.3.ab_ah_g_m_as $12$ (not in LMFDB) 5.3.ab_c_ad_d_as $12$ (not in LMFDB) 5.3.b_ah_ag_m_s $12$ (not in LMFDB) 5.3.b_c_d_d_s $12$ (not in LMFDB) 5.3.e_f_a_ay_acu $12$ (not in LMFDB) 5.3.h_r_g_aci_agg $12$ (not in LMFDB) 5.3.h_ba_cr_fr_kk $12$ (not in LMFDB) 5.3.e_i_d_abb_acu $18$ (not in LMFDB) 5.3.e_i_v_bt_cu $18$ (not in LMFDB) 5.3.an_de_amp_biz_acsq $24$ (not in LMFDB) 5.3.al_cd_agm_of_azz $24$ (not in LMFDB) 5.3.aj_bi_acl_bt_a $24$ (not in LMFDB) 5.3.aj_bm_adv_hh_amm $24$ (not in LMFDB) 5.3.ai_bf_adg_hh_anw $24$ (not in LMFDB) 5.3.ah_t_am_aco_hq $24$ (not in LMFDB) 5.3.ah_t_am_acl_hh $24$ (not in LMFDB) 5.3.ah_w_abh_j_bk $24$ (not in LMFDB) 5.3.ah_x_abw_da_aew $24$ (not in LMFDB) 5.3.ah_z_acc_dg_aew $24$ (not in LMFDB) 5.3.ah_bc_acx_gd_alc $24$ (not in LMFDB) 5.3.ah_bf_ads_ja_ari $24$ (not in LMFDB) 5.3.af_e_p_av_a $24$ (not in LMFDB) 5.3.af_h_a_j_abt $24$ (not in LMFDB) 5.3.af_k_ap_bn_adm $24$ (not in LMFDB) 5.3.af_k_ap_bt_aee $24$ (not in LMFDB) 5.3.af_n_abe_cr_aff $24$ (not in LMFDB) 5.3.af_q_abt_dv_agy $24$ (not in LMFDB) 5.3.ae_h_a_abe_cu $24$ (not in LMFDB) 5.3.ae_h_a_abb_cu $24$ (not in LMFDB) 5.3.ae_l_ay_bq_acu $24$ (not in LMFDB) 5.3.ae_n_ay_bw_acu $24$ (not in LMFDB) 5.3.ae_t_abw_ew_aii $24$ (not in LMFDB) 5.3.ad_af_y_g_adm $24$ (not in LMFDB) 5.3.ad_ac_p_j_acu $24$ (not in LMFDB) 5.3.ad_ab_m_ag_as $24$ (not in LMFDB) 5.3.ad_b_g_m_acc $24$ (not in LMFDB) 5.3.ad_c_d_j_abk $24$ (not in LMFDB) 5.3.ad_e_ad_p_abk $24$ (not in LMFDB) 5.3.ad_f_ag_y_acc $24$ (not in LMFDB) 5.3.ad_h_am_s_as $24$ (not in LMFDB) 5.3.ad_i_ap_bn_acu $24$ (not in LMFDB) 5.3.ad_l_ay_cc_adm $24$ (not in LMFDB) 5.3.ac_ac_a_ad_bk $24$ (not in LMFDB) 5.3.ac_b_ag_j_a $24$ (not in LMFDB) 5.3.ac_e_am_v_abk $24$ (not in LMFDB) 5.3.ac_h_as_bh_acu $24$ (not in LMFDB) 5.3.ac_k_ay_bt_aee $24$ (not in LMFDB) 5.3.ab_ai_p_p_acu $24$ (not in LMFDB) 5.3.ab_af_m_g_acc $24$ (not in LMFDB) 5.3.ab_af_m_j_abt $24$ (not in LMFDB) 5.3.ab_ac_j_d_as $24$ (not in LMFDB) 5.3.ab_ac_j_j_abk $24$ (not in LMFDB) 5.3.ab_ab_a_g_as $24$ (not in LMFDB) 5.3.ab_b_g_ad_j $24$ (not in LMFDB) 5.3.ab_b_g_m_as $24$ (not in LMFDB) 5.3.ab_e_d_aj_bk $24$ (not in LMFDB) 5.3.ab_e_d_p_a $24$ (not in LMFDB) 5.3.ab_h_a_s_s $24$ (not in LMFDB) 5.3.a_af_a_g_a $24$ (not in LMFDB) 5.3.a_ab_a_ag_a $24$ (not in LMFDB) 5.3.a_b_a_m_a $24$ (not in LMFDB) 5.3.a_f_a_y_a $24$ (not in LMFDB) 5.3.a_h_a_s_a $24$ (not in LMFDB) 5.3.a_l_a_cc_a $24$ (not in LMFDB) 5.3.b_ai_ap_p_cu $24$ (not in LMFDB) 5.3.b_af_am_g_cc $24$ (not in LMFDB) 5.3.b_af_am_j_bt $24$ (not in LMFDB) 5.3.b_ac_aj_d_s $24$ (not in LMFDB) 5.3.b_ac_aj_j_bk $24$ (not in LMFDB) 5.3.b_ab_a_g_s $24$ (not in LMFDB) 5.3.b_b_ag_ad_aj $24$ (not in LMFDB) 5.3.b_b_ag_m_s $24$ (not in LMFDB) 5.3.b_e_ad_aj_abk $24$ (not in LMFDB) 5.3.b_e_ad_p_a $24$ (not in LMFDB) 5.3.b_h_a_s_as $24$ (not in LMFDB) 5.3.c_ac_a_ad_abk $24$ (not in LMFDB) 5.3.c_b_g_j_a $24$ (not in LMFDB) 5.3.c_e_m_v_bk $24$ (not in LMFDB) 5.3.c_h_s_bh_cu $24$ (not in LMFDB) 5.3.c_k_y_bt_ee $24$ (not in LMFDB) 5.3.d_af_ay_g_dm $24$ (not in LMFDB) 5.3.d_ac_ap_j_cu $24$ (not in LMFDB) 5.3.d_ab_am_ag_s $24$ (not in LMFDB) 5.3.d_b_ag_m_cc $24$ (not in LMFDB) 5.3.d_c_ad_j_bk $24$ (not in LMFDB) 5.3.d_e_d_p_bk $24$ (not in LMFDB) 5.3.d_f_g_y_cc $24$ (not in LMFDB) 5.3.d_h_m_s_s $24$ (not in LMFDB) 5.3.d_i_p_bn_cu $24$ (not in LMFDB) 5.3.d_l_y_cc_dm $24$ (not in LMFDB) 5.3.e_h_a_abe_acu $24$ (not in LMFDB) 5.3.e_h_a_abb_acu $24$ (not in LMFDB) 5.3.e_l_y_bq_cu $24$ (not in LMFDB) 5.3.e_n_y_bw_cu $24$ (not in LMFDB) 5.3.e_t_bw_ew_ii $24$ (not in LMFDB) 5.3.f_e_ap_av_a $24$ (not in LMFDB) 5.3.f_h_a_j_bt $24$ (not in LMFDB) 5.3.f_k_p_bn_dm $24$ (not in LMFDB) 5.3.f_k_p_bt_ee $24$ (not in LMFDB) 5.3.f_n_be_cr_ff $24$ (not in LMFDB) 5.3.f_q_bt_dv_gy $24$ (not in LMFDB) 5.3.h_t_m_aco_ahq $24$ (not in LMFDB) 5.3.h_t_m_acl_ahh $24$ (not in LMFDB) 5.3.h_w_bh_j_abk $24$ (not in LMFDB) 5.3.h_x_bw_da_ew $24$ (not in LMFDB) 5.3.h_z_cc_dg_ew $24$ (not in LMFDB) 5.3.h_bc_cx_gd_lc $24$ (not in LMFDB) 5.3.h_bf_ds_ja_ri $24$ (not in LMFDB) 5.3.i_bf_dg_hh_nw $24$ (not in LMFDB) 5.3.j_bi_cl_bt_a $24$ (not in LMFDB) 5.3.j_bm_dv_hh_mm $24$ (not in LMFDB) 5.3.l_cd_gm_of_zz $24$ (not in LMFDB) 5.3.n_de_mp_biz_csq $24$ (not in LMFDB) 5.3.ae_k_av_bt_adm $72$ (not in LMFDB) 5.3.ae_k_ad_abb_dm $72$ (not in LMFDB) 5.3.ac_b_ap_bb_aj $72$ (not in LMFDB) 5.3.ac_b_d_aj_j $72$ (not in LMFDB) 5.3.a_ac_aj_j_s $72$ (not in LMFDB) 5.3.a_ac_j_j_as $72$ (not in LMFDB) 5.3.a_c_aj_j_as $72$ (not in LMFDB) 5.3.a_c_j_j_s $72$ (not in LMFDB) 5.3.c_b_ad_aj_aj $72$ (not in LMFDB) 5.3.c_b_p_bb_j $72$ (not in LMFDB) 5.3.e_k_d_abb_adm $72$ (not in LMFDB) 5.3.e_k_v_bt_dm $72$ (not in LMFDB)