# Properties

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

## Invariants

 Base field: $\F_{3}$ Dimension: $5$ L-polynomial: $( 1 - x + 3 x^{2} )( 1 - 3 x + 3 x^{2} )^{2}( 1 - 4 x + 8 x^{2} - 12 x^{3} + 9 x^{4} )$ Frobenius angles: $\pm0.0540867239847$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.406785250661$, $\pm0.445913276015$ Angle rank: $2$ (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 $A(\F_{q^r})$ 6 49980 17668224 2871850800 770915024106 235482089472000 57268208325165438 12552784305636556800 2891137306844780364672 706333278628119027939900

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -7 9 32 65 223 828 2485 6769 19256 58089

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ad 2 $\times$ 1.3.ab $\times$ 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.ab : $$\Q(\sqrt{-11})$$. 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 2 $\times$ 1.531441.zi 2 $\times$ 1.531441.cag. The endomorphism algebra for each factor is: 1.531441.acec 2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over $$\Q$$ ramified at $3$ and $\infty$. 1.531441.zi 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-2})$$$)$ 1.531441.cag : $$\Q(\sqrt{-11})$$.
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.f $\times$ 2.9.a_ao. The endomorphism algebra for each factor is: 1.9.ad 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$ 1.9.f : $$\Q(\sqrt{-11})$$. 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 2 $\times$ 1.27.i $\times$ 2.27.ae_i. The endomorphism algebra for each factor is: 1.27.a 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$ 1.27.i : $$\Q(\sqrt{-11})$$. 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.ao 2 $\times$ 1.81.ah $\times$ 1.81.j 2 . The endomorphism algebra for each factor is: 1.81.ao 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-2})$$$)$ 1.81.ah : $$\Q(\sqrt{-11})$$. 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.ak $\times$ 1.729.cc 2 $\times$ 2.729.a_zi. The endomorphism algebra for each factor is: 1.729.ak : $$\Q(\sqrt{-11})$$. 1.729.cc 2 : $\mathrm{M}_{2}(B)$, 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.aj_bo_aer_kz_ava $2$ (not in LMFDB) 5.3.ad_e_ah_j_ag $2$ (not in LMFDB) 5.3.ad_e_b_ap_bq $2$ (not in LMFDB) 5.3.ab_a_ab_ad_be $2$ (not in LMFDB) 5.3.d_e_ab_ap_abq $2$ (not in LMFDB) 5.3.d_e_h_j_g $2$ (not in LMFDB) 5.3.f_m_r_j_ag $2$ (not in LMFDB) 5.3.j_bo_er_kz_va $2$ (not in LMFDB) 5.3.l_ci_ih_vp_bqs $2$ (not in LMFDB) 5.3.ai_bk_aem_kz_avg $3$ (not in LMFDB) 5.3.af_m_ar_j_g $3$ (not in LMFDB) 5.3.af_v_ack_fo_akw $3$ (not in LMFDB) 5.3.ac_g_ai_d_am $3$ (not in LMFDB) 5.3.b_a_b_ad_abe $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 5.3.aj_bo_aer_kz_ava $2$ (not in LMFDB) 5.3.ad_e_ah_j_ag $2$ (not in LMFDB) 5.3.ad_e_b_ap_bq $2$ (not in LMFDB) 5.3.ab_a_ab_ad_be $2$ (not in LMFDB) 5.3.d_e_ab_ap_abq $2$ (not in LMFDB) 5.3.d_e_h_j_g $2$ (not in LMFDB) 5.3.f_m_r_j_ag $2$ (not in LMFDB) 5.3.j_bo_er_kz_va $2$ (not in LMFDB) 5.3.l_ci_ih_vp_bqs $2$ (not in LMFDB) 5.3.ai_bk_aem_kz_avg $3$ (not in LMFDB) 5.3.af_m_ar_j_g $3$ (not in LMFDB) 5.3.af_v_ack_fo_akw $3$ (not in LMFDB) 5.3.ac_g_ai_d_am $3$ (not in LMFDB) 5.3.b_a_b_ad_abe $3$ (not in LMFDB) 5.3.af_s_abv_dv_ahe $4$ (not in LMFDB) 5.3.ad_k_az_bz_ady $4$ (not in LMFDB) 5.3.d_k_z_bz_dy $4$ (not in LMFDB) 5.3.f_s_bv_dv_he $4$ (not in LMFDB) 5.3.ai_bk_aem_kz_avg $6$ (not in LMFDB) 5.3.ag_w_acm_fr_akq $6$ (not in LMFDB) 5.3.ad_n_abi_cu_afu $6$ (not in LMFDB) 5.3.a_e_ae_ad_ay $6$ (not in LMFDB) 5.3.a_e_e_ad_y $6$ (not in LMFDB) 5.3.c_g_i_d_m $6$ (not in LMFDB) 5.3.d_n_bi_cu_fu $6$ (not in LMFDB) 5.3.f_v_ck_fo_kw $6$ (not in LMFDB) 5.3.g_w_cm_fr_kq $6$ (not in LMFDB) 5.3.i_bk_em_kz_vg $6$ (not in LMFDB) 5.3.al_ck_aiv_xl_abuq $8$ (not in LMFDB) 5.3.aj_bq_afb_lx_awq $8$ (not in LMFDB) 5.3.ah_w_abl_bh_ay $8$ (not in LMFDB) 5.3.ah_ba_acn_ez_aiu $8$ (not in LMFDB) 5.3.af_k_al_v_abw $8$ (not in LMFDB) 5.3.af_o_abf_cr_afc $8$ (not in LMFDB) 5.3.af_o_at_j_m $8$ (not in LMFDB) 5.3.af_u_abx_eh_ahk $8$ (not in LMFDB) 5.3.ad_g_an_bh_aci $8$ (not in LMFDB) 5.3.ad_g_af_j_am $8$ (not in LMFDB) 5.3.ad_m_ax_cl_ads $8$ (not in LMFDB) 5.3.ab_ac_f_j_ay $8$ (not in LMFDB) 5.3.ab_c_al_v_am $8$ (not in LMFDB) 5.3.ab_c_b_j_am $8$ (not in LMFDB) 5.3.ab_e_ab_p_am $8$ (not in LMFDB) 5.3.ab_i_af_bn_ay $8$ (not in LMFDB) 5.3.b_ac_af_j_y $8$ (not in LMFDB) 5.3.b_c_ab_j_m $8$ (not in LMFDB) 5.3.b_c_l_v_m $8$ (not in LMFDB) 5.3.b_e_b_p_m $8$ (not in LMFDB) 5.3.b_i_f_bn_y $8$ (not in LMFDB) 5.3.d_g_f_j_m $8$ (not in LMFDB) 5.3.d_g_n_bh_ci $8$ (not in LMFDB) 5.3.d_m_x_cl_ds $8$ (not in LMFDB) 5.3.f_k_l_v_bw $8$ (not in LMFDB) 5.3.f_o_t_j_am $8$ (not in LMFDB) 5.3.f_o_bf_cr_fc $8$ (not in LMFDB) 5.3.f_u_bx_eh_hk $8$ (not in LMFDB) 5.3.h_w_bl_bh_y $8$ (not in LMFDB) 5.3.h_ba_cn_ez_iu $8$ (not in LMFDB) 5.3.j_bq_fb_lx_wq $8$ (not in LMFDB) 5.3.l_ck_iv_xl_buq $8$ (not in LMFDB) 5.3.af_j_ac_abk_dy $12$ (not in LMFDB) 5.3.ad_b_c_am_bq $12$ (not in LMFDB) 5.3.d_b_ac_am_abq $12$ (not in LMFDB) 5.3.f_j_c_abk_ady $12$ (not in LMFDB) 5.3.aj_bn_aei_jp_arx $24$ (not in LMFDB) 5.3.ai_bm_aeu_lx_axc $24$ (not in LMFDB) 5.3.ah_x_ace_ez_ajv $24$ (not in LMFDB) 5.3.ag_v_acg_ez_ajg $24$ (not in LMFDB) 5.3.ag_y_acq_gd_alo $24$ (not in LMFDB) 5.3.af_l_ae_abq_ek $24$ (not in LMFDB) 5.3.af_l_ae_abn_eh $24$ (not in LMFDB) 5.3.af_p_abg_cc_adm $24$ (not in LMFDB) 5.3.af_r_abi_ci_adm $24$ (not in LMFDB) 5.3.af_x_acm_gg_ali $24$ (not in LMFDB) 5.3.ae_k_aq_v_ay $24$ (not in LMFDB) 5.3.ae_l_abg_cr_aeq $24$ (not in LMFDB) 5.3.ae_o_abg_cr_aeq $24$ (not in LMFDB) 5.3.ad_a_f_aj_y $24$ (not in LMFDB) 5.3.ad_d_ae_j_ap $24$ (not in LMFDB) 5.3.ad_d_e_as_be $24$ (not in LMFDB) 5.3.ad_d_e_ap_bh $24$ (not in LMFDB) 5.3.ad_g_an_bb_acc $24$ (not in LMFDB) 5.3.ad_h_aq_be_acc $24$ (not in LMFDB) 5.3.ad_j_aw_bt_adp $24$ (not in LMFDB) 5.3.ad_j_ao_bk_acc $24$ (not in LMFDB) 5.3.ad_m_abf_cl_afc $24$ (not in LMFDB) 5.3.ad_p_abg_dm_afi $24$ (not in LMFDB) 5.3.ac_e_ai_p_am $24$ (not in LMFDB) 5.3.ac_f_c_ap_bw $24$ (not in LMFDB) 5.3.ac_i_aq_bn_aci $24$ (not in LMFDB) 5.3.ac_i_ae_p_m $24$ (not in LMFDB) 5.3.ab_af_i_g_abe $24$ (not in LMFDB) 5.3.ab_ae_af_d_bw $24$ (not in LMFDB) 5.3.ab_ab_ai_j_p $24$ (not in LMFDB) 5.3.ab_ab_e_ag_ag $24$ (not in LMFDB) 5.3.ab_b_c_m_as $24$ (not in LMFDB) 5.3.ab_c_al_p_as $24$ (not in LMFDB) 5.3.ab_f_ao_v_abz $24$ (not in LMFDB) 5.3.ab_f_ac_y_as $24$ (not in LMFDB) 5.3.ab_h_ae_s_ag $24$ (not in LMFDB) 5.3.ab_i_ar_bb_adg $24$ (not in LMFDB) 5.3.ab_l_ai_cc_abe $24$ (not in LMFDB) 5.3.a_d_ae_ad_ay $24$ (not in LMFDB) 5.3.a_d_e_ad_y $24$ (not in LMFDB) 5.3.a_g_ae_v_ay $24$ (not in LMFDB) 5.3.a_g_e_v_y $24$ (not in LMFDB) 5.3.b_af_ai_g_be $24$ (not in LMFDB) 5.3.b_ae_f_d_abw $24$ (not in LMFDB) 5.3.b_ab_ae_ag_g $24$ (not in LMFDB) 5.3.b_ab_i_j_ap $24$ (not in LMFDB) 5.3.b_b_ac_m_s $24$ (not in LMFDB) 5.3.b_c_l_p_s $24$ (not in LMFDB) 5.3.b_f_c_y_s $24$ (not in LMFDB) 5.3.b_f_o_v_bz $24$ (not in LMFDB) 5.3.b_h_e_s_g $24$ (not in LMFDB) 5.3.b_i_r_bb_dg $24$ (not in LMFDB) 5.3.b_l_i_cc_be $24$ (not in LMFDB) 5.3.c_e_i_p_m $24$ (not in LMFDB) 5.3.c_f_ac_ap_abw $24$ (not in LMFDB) 5.3.c_i_e_p_am $24$ (not in LMFDB) 5.3.c_i_q_bn_ci $24$ (not in LMFDB) 5.3.d_a_af_aj_ay $24$ (not in LMFDB) 5.3.d_d_ae_as_abe $24$ (not in LMFDB) 5.3.d_d_ae_ap_abh $24$ (not in LMFDB) 5.3.d_d_e_j_p $24$ (not in LMFDB) 5.3.d_g_n_bb_cc $24$ (not in LMFDB) 5.3.d_h_q_be_cc $24$ (not in LMFDB) 5.3.d_j_o_bk_cc $24$ (not in LMFDB) 5.3.d_j_w_bt_dp $24$ (not in LMFDB) 5.3.d_m_bf_cl_fc $24$ (not in LMFDB) 5.3.d_p_bg_dm_fi $24$ (not in LMFDB) 5.3.e_k_q_v_y $24$ (not in LMFDB) 5.3.e_l_bg_cr_eq $24$ (not in LMFDB) 5.3.e_o_bg_cr_eq $24$ (not in LMFDB) 5.3.f_l_e_abq_aek $24$ (not in LMFDB) 5.3.f_l_e_abn_aeh $24$ (not in LMFDB) 5.3.f_p_bg_cc_dm $24$ (not in LMFDB) 5.3.f_r_bi_ci_dm $24$ (not in LMFDB) 5.3.f_x_cm_gg_li $24$ (not in LMFDB) 5.3.g_v_cg_ez_jg $24$ (not in LMFDB) 5.3.g_y_cq_gd_lo $24$ (not in LMFDB) 5.3.h_x_ce_ez_jv $24$ (not in LMFDB) 5.3.i_bm_eu_lx_xc $24$ (not in LMFDB) 5.3.j_bn_ei_jp_rx $24$ (not in LMFDB)