Properties

 Label 5.3.an_de_amp_biz_acsq 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: $( 1 - 2 x + 3 x^{2} )^{2}( 1 - 3 x + 3 x^{2} )^{3}$ Frobenius angles: $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.304086723985$, $\pm0.304086723985$ 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 $A(\F_{q^r})$ 4 49392 31698688 6944910336 1165570654204 225455270068224 51614165618530036 12492606163646988288 3012386135220532021504 720755591632702229851632

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -9 5 48 137 321 800 2259 6737 20064 59285

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ad 3 $\times$ 1.3.ac 2 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 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-3})$$$)$ 1.3.ac 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-2})$$$)$
Endomorphism algebra over $\overline{\F}_{3}$
 The base change of $A$ to $\F_{3^{6}}$ is 1.729.abu 2 $\times$ 1.729.cc 3 . The endomorphism algebra for each factor is: 1.729.abu 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-2})$$$)$ 1.729.cc 3 : $\mathrm{M}_{3}(B)$, where $B$ is the quaternion algebra over $$\Q$$ ramified at $3$ and $\infty$.
All geometric endomorphisms are defined over $\F_{3^{6}}$.
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 3 $\times$ 1.9.c 2 . The endomorphism algebra for each factor is: 1.9.ad 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-3})$$$)$ 1.9.c 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-2})$$$)$
• Endomorphism algebra over $\F_{3^{3}}$  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a 3 $\times$ 1.27.k 2 . The endomorphism algebra for each factor is: 1.27.a 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-3})$$$)$ 1.27.k 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-2})$$$)$

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.aj_bm_adv_hh_amm $2$ (not in LMFDB) 5.3.ah_w_abh_j_bk $2$ (not in LMFDB) 5.3.af_k_ap_bt_aee $2$ (not in LMFDB) 5.3.ad_c_d_j_abk $2$ (not in LMFDB) 5.3.ab_ac_j_j_abk $2$ (not in LMFDB) 5.3.b_ac_aj_j_bk $2$ (not in LMFDB) 5.3.d_c_ad_j_bk $2$ (not in LMFDB) 5.3.f_k_p_bt_ee $2$ (not in LMFDB) 5.3.h_w_bh_j_abk $2$ (not in LMFDB) 5.3.j_bm_dv_hh_mm $2$ (not in LMFDB) 5.3.n_de_mp_biz_csq $2$ (not in LMFDB) 5.3.ak_ca_agy_rr_abiq $3$ (not in LMFDB) 5.3.ah_t_am_acl_hh $3$ (not in LMFDB) 5.3.ah_w_abh_j_bk $3$ (not in LMFDB) 5.3.ah_bf_ads_ja_ari $3$ (not in LMFDB) 5.3.ae_h_a_abb_cu $3$ (not in LMFDB) 5.3.ae_k_am_j_a $3$ (not in LMFDB) 5.3.ae_t_abw_ew_aii $3$ (not in LMFDB) 5.3.ab_af_m_j_abt $3$ (not in LMFDB) 5.3.ab_ac_j_j_abk $3$ (not in LMFDB) 5.3.ab_e_d_aj_bk $3$ (not in LMFDB) 5.3.ab_h_a_s_s $3$ (not in LMFDB) 5.3.c_b_g_j_a $3$ (not in LMFDB) 5.3.c_e_m_bb_bk $3$ (not in LMFDB) 5.3.c_k_y_bt_ee $3$ (not in LMFDB) 5.3.f_h_a_j_bt $3$ (not in LMFDB) 5.3.f_k_p_bt_ee $3$ (not in LMFDB) 5.3.f_q_bt_dv_gy $3$ (not in LMFDB) 5.3.i_bf_dg_hh_nw $3$ (not in LMFDB) 5.3.l_cd_gm_of_zz $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 5.3.aj_bm_adv_hh_amm $2$ (not in LMFDB) 5.3.ah_w_abh_j_bk $2$ (not in LMFDB) 5.3.af_k_ap_bt_aee $2$ (not in LMFDB) 5.3.ad_c_d_j_abk $2$ (not in LMFDB) 5.3.ab_ac_j_j_abk $2$ (not in LMFDB) 5.3.b_ac_aj_j_bk $2$ (not in LMFDB) 5.3.d_c_ad_j_bk $2$ (not in LMFDB) 5.3.f_k_p_bt_ee $2$ (not in LMFDB) 5.3.h_w_bh_j_abk $2$ (not in LMFDB) 5.3.j_bm_dv_hh_mm $2$ (not in LMFDB) 5.3.n_de_mp_biz_csq $2$ (not in LMFDB) 5.3.ak_ca_agy_rr_abiq $3$ (not in LMFDB) 5.3.ah_t_am_acl_hh $3$ (not in LMFDB) 5.3.ah_w_abh_j_bk $3$ (not in LMFDB) 5.3.ah_bf_ads_ja_ari $3$ (not in LMFDB) 5.3.ae_h_a_abb_cu $3$ (not in LMFDB) 5.3.ae_k_am_j_a $3$ (not in LMFDB) 5.3.ae_t_abw_ew_aii $3$ (not in LMFDB) 5.3.ab_af_m_j_abt $3$ (not in LMFDB) 5.3.ab_ac_j_j_abk $3$ (not in LMFDB) 5.3.ab_e_d_aj_bk $3$ (not in LMFDB) 5.3.ab_h_a_s_s $3$ (not in LMFDB) 5.3.c_b_g_j_a $3$ (not in LMFDB) 5.3.c_e_m_bb_bk $3$ (not in LMFDB) 5.3.c_k_y_bt_ee $3$ (not in LMFDB) 5.3.f_h_a_j_bt $3$ (not in LMFDB) 5.3.f_k_p_bt_ee $3$ (not in LMFDB) 5.3.f_q_bt_dv_gy $3$ (not in LMFDB) 5.3.i_bf_dg_hh_nw $3$ (not in LMFDB) 5.3.l_cd_gm_of_zz $3$ (not in LMFDB) 5.3.aj_bi_acl_bt_a $4$ (not in LMFDB) 5.3.ah_bc_acx_gd_alc $4$ (not in LMFDB) 5.3.ad_ac_p_j_acu $4$ (not in LMFDB) 5.3.ad_e_ad_p_abk $4$ (not in LMFDB) 5.3.ad_i_ap_bn_acu $4$ (not in LMFDB) 5.3.ab_e_d_p_a $4$ (not in LMFDB) 5.3.b_e_ad_p_a $4$ (not in LMFDB) 5.3.d_ac_ap_j_cu $4$ (not in LMFDB) 5.3.d_e_d_p_bk $4$ (not in LMFDB) 5.3.d_i_p_bn_cu $4$ (not in LMFDB) 5.3.h_bc_cx_gd_lc $4$ (not in LMFDB) 5.3.j_bi_cl_bt_a $4$ (not in LMFDB) 5.3.al_cd_agm_of_azz $6$ (not in LMFDB) 5.3.ai_bf_adg_hh_anw $6$ (not in LMFDB) 5.3.ag_u_abw_dv_agy $6$ (not in LMFDB) 5.3.af_h_a_j_abt $6$ (not in LMFDB) 5.3.af_q_abt_dv_agy $6$ (not in LMFDB) 5.3.ad_l_ay_cc_adm $6$ (not in LMFDB) 5.3.ac_b_ag_j_a $6$ (not in LMFDB) 5.3.ac_e_am_bb_abk $6$ (not in LMFDB) 5.3.ac_k_ay_bt_aee $6$ (not in LMFDB) 5.3.a_c_a_j_a $6$ (not in LMFDB) 5.3.a_l_a_cc_a $6$ (not in LMFDB) 5.3.b_af_am_j_bt $6$ (not in LMFDB) 5.3.b_e_ad_aj_abk $6$ (not in LMFDB) 5.3.b_h_a_s_as $6$ (not in LMFDB) 5.3.d_l_y_cc_dm $6$ (not in LMFDB) 5.3.e_h_a_abb_acu $6$ (not in LMFDB) 5.3.e_k_m_j_a $6$ (not in LMFDB) 5.3.e_t_bw_ew_ii $6$ (not in LMFDB) 5.3.g_u_bw_dv_gy $6$ (not in LMFDB) 5.3.h_t_m_acl_ahh $6$ (not in LMFDB) 5.3.h_bf_ds_ja_ri $6$ (not in LMFDB) 5.3.k_ca_gy_rr_biq $6$ (not in LMFDB) 5.3.an_dc_alx_bgf_acmk $8$ (not in LMFDB) 5.3.ah_u_abb_j_s $8$ (not in LMFDB) 5.3.ah_ba_acr_fr_akk $8$ (not in LMFDB) 5.3.af_i_d_abb_cc $8$ (not in LMFDB) 5.3.ab_ae_d_j_as $8$ (not in LMFDB) 5.3.ab_c_ad_d_as $8$ (not in LMFDB) 5.3.b_ae_ad_j_s $8$ (not in LMFDB) 5.3.b_c_d_d_s $8$ (not in LMFDB) 5.3.f_i_ad_abb_acc $8$ (not in LMFDB) 5.3.h_u_bb_j_as $8$ (not in LMFDB) 5.3.h_ba_cr_fr_kk $8$ (not in LMFDB) 5.3.n_dc_lx_bgf_cmk $8$ (not in LMFDB) 5.3.ae_k_av_bt_adm $9$ (not in LMFDB) 5.3.ae_k_ad_abb_dm $9$ (not in LMFDB) 5.3.c_b_ad_aj_aj $9$ (not in LMFDB) 5.3.c_b_p_bb_j $9$ (not in LMFDB) 5.3.ah_t_am_aco_hq $12$ (not in LMFDB) 5.3.ag_q_ay_bb_abk $12$ (not in LMFDB) 5.3.af_e_p_av_a $12$ (not in LMFDB) 5.3.af_n_abe_cr_aff $12$ (not in LMFDB) 5.3.ae_h_a_abe_cu $12$ (not in LMFDB) 5.3.ae_q_abk_dj_afo $12$ (not in LMFDB) 5.3.ad_af_y_g_adm $12$ (not in LMFDB) 5.3.ad_ab_m_ag_as $12$ (not in LMFDB) 5.3.ad_h_am_s_as $12$ (not in LMFDB) 5.3.ac_ac_a_ad_bk $12$ (not in LMFDB) 5.3.ac_h_as_bh_acu $12$ (not in LMFDB) 5.3.ab_ai_p_p_acu $12$ (not in LMFDB) 5.3.ab_af_m_g_acc $12$ (not in LMFDB) 5.3.ab_b_g_ad_j $12$ (not in LMFDB) 5.3.a_af_a_g_a $12$ (not in LMFDB) 5.3.a_ac_a_j_a $12$ (not in LMFDB) 5.3.a_ab_a_ag_a $12$ (not in LMFDB) 5.3.a_e_a_p_a $12$ (not in LMFDB) 5.3.a_h_a_s_a $12$ (not in LMFDB) 5.3.a_i_a_bn_a $12$ (not in LMFDB) 5.3.b_ai_ap_p_cu $12$ (not in LMFDB) 5.3.b_af_am_g_cc $12$ (not in LMFDB) 5.3.b_b_ag_ad_aj $12$ (not in LMFDB) 5.3.c_ac_a_ad_abk $12$ (not in LMFDB) 5.3.c_h_s_bh_cu $12$ (not in LMFDB) 5.3.d_af_ay_g_dm $12$ (not in LMFDB) 5.3.d_ab_am_ag_s $12$ (not in LMFDB) 5.3.d_h_m_s_s $12$ (not in LMFDB) 5.3.e_h_a_abe_acu $12$ (not in LMFDB) 5.3.e_q_bk_dj_fo $12$ (not in LMFDB) 5.3.f_e_ap_av_a $12$ (not in LMFDB) 5.3.f_n_be_cr_ff $12$ (not in LMFDB) 5.3.g_q_y_bb_bk $12$ (not in LMFDB) 5.3.h_t_m_aco_ahq $12$ (not in LMFDB) 5.3.ac_b_ap_bb_aj $18$ (not in LMFDB) 5.3.ac_b_d_aj_j $18$ (not in LMFDB) 5.3.a_c_aj_j_as $18$ (not in LMFDB) 5.3.a_c_j_j_s $18$ (not in LMFDB) 5.3.e_k_d_abb_adm $18$ (not in LMFDB) 5.3.e_k_v_bt_dm $18$ (not in LMFDB) 5.3.ak_by_agm_qh_abfw $24$ (not in LMFDB) 5.3.ah_r_ag_aci_gg $24$ (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_bd_adm_ii_apy $24$ (not in LMFDB) 5.3.af_k_ap_bn_adm $24$ (not in LMFDB) 5.3.ae_f_a_ay_cu $24$ (not in LMFDB) 5.3.ae_i_am_j_a $24$ (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_o_abk_cx_afo $24$ (not in LMFDB) 5.3.ae_r_abw_ee_aii $24$ (not in LMFDB) 5.3.ad_b_g_m_acc $24$ (not in LMFDB) 5.3.ad_f_ag_y_acc $24$ (not in LMFDB) 5.3.ac_c_a_aj_bk $24$ (not in LMFDB) 5.3.ac_e_am_v_abk $24$ (not in LMFDB) 5.3.ab_ah_g_m_as $24$ (not in LMFDB) 5.3.ab_ac_j_d_as $24$ (not in LMFDB) 5.3.ab_ab_a_g_as $24$ (not in LMFDB) 5.3.ab_b_g_m_as $24$ (not in LMFDB) 5.3.ab_f_ag_a_as $24$ (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.b_ah_ag_m_s $24$ (not in LMFDB) 5.3.b_ac_aj_d_s $24$ (not in LMFDB) 5.3.b_ab_a_g_s $24$ (not in LMFDB) 5.3.b_b_ag_m_s $24$ (not in LMFDB) 5.3.b_f_g_a_s $24$ (not in LMFDB) 5.3.c_c_a_aj_abk $24$ (not in LMFDB) 5.3.c_e_m_v_bk $24$ (not in LMFDB) 5.3.d_b_ag_m_cc $24$ (not in LMFDB) 5.3.d_f_g_y_cc $24$ (not in LMFDB) 5.3.e_f_a_ay_acu $24$ (not in LMFDB) 5.3.e_i_m_j_a $24$ (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_o_bk_cx_fo $24$ (not in LMFDB) 5.3.e_r_bw_ee_ii $24$ (not in LMFDB) 5.3.f_k_p_bn_dm $24$ (not in LMFDB) 5.3.h_r_g_aci_agg $24$ (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_bd_dm_ii_py $24$ (not in LMFDB) 5.3.k_by_gm_qh_bfw $24$ (not in LMFDB) 5.3.a_ac_aj_j_s $36$ (not in LMFDB) 5.3.a_ac_j_j_as $36$ (not in LMFDB) 5.3.ae_i_av_bt_acu $72$ (not in LMFDB) 5.3.ae_i_ad_abb_cu $72$ (not in LMFDB) 5.3.e_i_d_abb_acu $72$ (not in LMFDB) 5.3.e_i_v_bt_cu $72$ (not in LMFDB)