# Properties

 Label 5.3.am_cs_akc_bax_acbo 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: $( 1 - 2 x + 3 x^{2} )( 1 - 3 x + 3 x^{2} )^{2}( 1 - 4 x + 8 x^{2} - 12 x^{3} + 9 x^{4} )$ Frobenius angles: $\pm0.0540867239847$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.304086723985$, $\pm0.445913276015$ 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 $A(\F_{q^r})$ 4 39984 18649792 3675969024 875875285604 223707984998400 53006505459928556 12276340965871976448 2939352508503845148736 717873371631057310649904

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -8 6 34 86 252 792 2316 6622 19582 59046

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ad 2 $\times$ 1.3.ac $\times$ 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.ac : $$\Q(\sqrt{-2})$$. 2.3.ae_i : $$\Q(\zeta_{8})$$.
Endomorphism algebra over $\overline{\F}_{3}$
 The base change of $A$ to $\F_{3^{24}}$ is 1.282429536481.acimic 2 $\times$ 1.282429536481.bjvvq 3 . The endomorphism algebra for each factor is: 1.282429536481.acimic 2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over $$\Q$$ ramified at $3$ and $\infty$. 1.282429536481.bjvvq 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-2})$$$)$
All geometric endomorphisms are defined over $\F_{3^{24}}$.
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.c $\times$ 2.9.a_ao. The endomorphism algebra for each factor is: 1.9.ad 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$ 1.9.c : $$\Q(\sqrt{-2})$$. 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.k $\times$ 2.27.ae_i. The endomorphism algebra for each factor is: 1.27.a 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$ 1.27.k : $$\Q(\sqrt{-2})$$. 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.j 2 $\times$ 1.81.o. The endomorphism algebra for each factor is: 1.81.ao 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-2})$$$)$ 1.81.j 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$ 1.81.o : $$\Q(\sqrt{-2})$$.
• Endomorphism algebra over $\F_{3^{6}}$  The base change of $A$ to $\F_{3^{6}}$ is 1.729.abu $\times$ 1.729.cc 2 $\times$ 2.729.a_zi. The endomorphism algebra for each factor is: 1.729.abu : $$\Q(\sqrt{-2})$$. 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 : $$\Q(\zeta_{8})$$.
• Endomorphism algebra over $\F_{3^{8}}$  The base change of $A$ to $\F_{3^{8}}$ is 1.6561.abi 3 $\times$ 1.6561.dd 2 . The endomorphism algebra for each factor is: 1.6561.abi 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-2})$$$)$ 1.6561.dd 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$
• Endomorphism algebra over $\F_{3^{12}}$  The base change of $A$ to $\F_{3^{12}}$ is 1.531441.acec 2 $\times$ 1.531441.azi $\times$ 1.531441.zi 2 . 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.azi : $$\Q(\sqrt{-2})$$. 1.531441.zi 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.ai_be_acw_fr_ake $2$ (not in LMFDB) 5.3.ag_q_aw_j_m $2$ (not in LMFDB) 5.3.ae_g_c_av_bw $2$ (not in LMFDB) 5.3.ac_a_ac_j_am $2$ (not in LMFDB) 5.3.a_ac_ac_d_y $2$ (not in LMFDB) 5.3.a_ac_c_d_ay $2$ (not in LMFDB) 5.3.c_a_c_j_m $2$ (not in LMFDB) 5.3.e_g_ac_av_abw $2$ (not in LMFDB) 5.3.g_q_w_j_am $2$ (not in LMFDB) 5.3.i_be_cw_fr_ke $2$ (not in LMFDB) 5.3.m_cs_kc_bax_cbo $2$ (not in LMFDB) 5.3.aj_br_afm_nq_abao $3$ (not in LMFDB) 5.3.ag_q_aw_j_m $3$ (not in LMFDB) 5.3.ag_z_acy_gy_ank $3$ (not in LMFDB) 5.3.ad_h_ak_g_ag $3$ (not in LMFDB) 5.3.a_ac_c_d_ay $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 5.3.ai_be_acw_fr_ake $2$ (not in LMFDB) 5.3.ag_q_aw_j_m $2$ (not in LMFDB) 5.3.ae_g_c_av_bw $2$ (not in LMFDB) 5.3.ac_a_ac_j_am $2$ (not in LMFDB) 5.3.a_ac_ac_d_y $2$ (not in LMFDB) 5.3.a_ac_c_d_ay $2$ (not in LMFDB) 5.3.c_a_c_j_m $2$ (not in LMFDB) 5.3.e_g_ac_av_abw $2$ (not in LMFDB) 5.3.g_q_w_j_am $2$ (not in LMFDB) 5.3.i_be_cw_fr_ke $2$ (not in LMFDB) 5.3.m_cs_kc_bax_cbo $2$ (not in LMFDB) 5.3.aj_br_afm_nq_abao $3$ (not in LMFDB) 5.3.ag_q_aw_j_m $3$ (not in LMFDB) 5.3.ag_z_acy_gy_ank $3$ (not in LMFDB) 5.3.ad_h_ak_g_ag $3$ (not in LMFDB) 5.3.a_ac_c_d_ay $3$ (not in LMFDB) 5.3.ag_w_acg_et_aiu $4$ (not in LMFDB) 5.3.ac_g_ao_bb_aci $4$ (not in LMFDB) 5.3.c_g_o_bb_ci $4$ (not in LMFDB) 5.3.g_w_cg_et_iu $4$ (not in LMFDB) 5.3.af_p_abm_da_afi $6$ (not in LMFDB) 5.3.ac_j_au_bk_adg $6$ (not in LMFDB) 5.3.ab_d_c_ag_be $6$ (not in LMFDB) 5.3.b_d_ac_ag_abe $6$ (not in LMFDB) 5.3.c_j_u_bk_dg $6$ (not in LMFDB) 5.3.d_h_k_g_g $6$ (not in LMFDB) 5.3.f_p_bm_da_fi $6$ (not in LMFDB) 5.3.g_z_cy_gy_nk $6$ (not in LMFDB) 5.3.j_br_fm_nq_bao $6$ (not in LMFDB) 5.3.am_cu_aks_bdf_acgq $8$ (not in LMFDB) 5.3.ai_bc_aby_bn_am $8$ (not in LMFDB) 5.3.ai_bg_ade_gd_akq $8$ (not in LMFDB) 5.3.ag_s_aba_j_y $8$ (not in LMFDB) 5.3.ag_y_ack_ff_ajg $8$ (not in LMFDB) 5.3.ae_e_c_p_aci $8$ (not in LMFDB) 5.3.ae_i_ao_bn_adg $8$ (not in LMFDB) 5.3.ac_ac_k_j_abw $8$ (not in LMFDB) 5.3.ac_c_c_j_ay $8$ (not in LMFDB) 5.3.ac_e_ac_p_ay $8$ (not in LMFDB) 5.3.ac_i_ak_bn_abw $8$ (not in LMFDB) 5.3.a_a_ak_p_m $8$ (not in LMFDB) 5.3.a_a_k_p_am $8$ (not in LMFDB) 5.3.c_ac_ak_j_bw $8$ (not in LMFDB) 5.3.c_c_ac_j_y $8$ (not in LMFDB) 5.3.c_e_c_p_y $8$ (not in LMFDB) 5.3.c_i_k_bn_bw $8$ (not in LMFDB) 5.3.e_e_ac_p_ci $8$ (not in LMFDB) 5.3.e_i_o_bn_dg $8$ (not in LMFDB) 5.3.g_s_ba_j_ay $8$ (not in LMFDB) 5.3.g_y_ck_ff_jg $8$ (not in LMFDB) 5.3.i_bc_by_bn_m $8$ (not in LMFDB) 5.3.i_bg_de_gd_kq $8$ (not in LMFDB) 5.3.m_cu_ks_bdf_cgq $8$ (not in LMFDB) 5.3.ag_n_ae_abw_fc $12$ (not in LMFDB) 5.3.ac_ad_e_a_m $12$ (not in LMFDB) 5.3.c_ad_ae_a_am $12$ (not in LMFDB) 5.3.g_n_e_abw_afc $12$ (not in LMFDB) 5.3.ak_bv_afk_lx_avy $24$ (not in LMFDB) 5.3.aj_bt_afw_ou_abcw $24$ (not in LMFDB) 5.3.ah_ba_act_gd_alo $24$ (not in LMFDB) 5.3.ag_p_abc_cr_afu $24$ (not in LMFDB) 5.3.ag_p_ai_acc_ga $24$ (not in LMFDB) 5.3.ag_p_ai_abz_fu $24$ (not in LMFDB) 5.3.ag_t_abo_co_aee $24$ (not in LMFDB) 5.3.ag_v_abs_cu_aee $24$ (not in LMFDB) 5.3.ag_bb_adc_hq_aoi $24$ (not in LMFDB) 5.3.af_n_au_y_abe $24$ (not in LMFDB) 5.3.af_r_abo_dg_afu $24$ (not in LMFDB) 5.3.ae_c_k_ap_m $24$ (not in LMFDB) 5.3.ae_f_ac_j_abe $24$ (not in LMFDB) 5.3.ae_i_ao_bh_acu $24$ (not in LMFDB) 5.3.ae_l_aba_cf_aek $24$ (not in LMFDB) 5.3.ae_o_abm_dd_aga $24$ (not in LMFDB) 5.3.ad_g_at_bn_aci $24$ (not in LMFDB) 5.3.ad_g_b_av_ci $24$ (not in LMFDB) 5.3.ad_j_ai_m_g $24$ (not in LMFDB) 5.3.ac_af_q_g_aci $24$ (not in LMFDB) 5.3.ac_ab_i_ag_am $24$ (not in LMFDB) 5.3.ac_ab_i_ad_ag $24$ (not in LMFDB) 5.3.ac_b_e_m_abk $24$ (not in LMFDB) 5.3.ac_d_ai_s_abk $24$ (not in LMFDB) 5.3.ac_f_ae_y_abk $24$ (not in LMFDB) 5.3.ac_h_ai_s_am $24$ (not in LMFDB) 5.3.ac_l_aq_cc_aci $24$ (not in LMFDB) 5.3.ab_b_ae_m_ag $24$ (not in LMFDB) 5.3.ab_c_af_d_am $24$ (not in LMFDB) 5.3.ab_f_ai_y_abe $24$ (not in LMFDB) 5.3.a_ag_ak_j_ci $24$ (not in LMFDB) 5.3.a_ag_k_j_aci $24$ (not in LMFDB) 5.3.a_ad_ak_j_be $24$ (not in LMFDB) 5.3.a_ad_k_j_abe $24$ (not in LMFDB) 5.3.a_a_ak_j_a $24$ (not in LMFDB) 5.3.a_a_k_j_a $24$ (not in LMFDB) 5.3.a_d_ak_j_abe $24$ (not in LMFDB) 5.3.a_d_k_j_be $24$ (not in LMFDB) 5.3.a_g_ak_j_aci $24$ (not in LMFDB) 5.3.a_g_k_j_ci $24$ (not in LMFDB) 5.3.b_b_e_m_g $24$ (not in LMFDB) 5.3.b_c_f_d_m $24$ (not in LMFDB) 5.3.b_f_i_y_be $24$ (not in LMFDB) 5.3.c_af_aq_g_ci $24$ (not in LMFDB) 5.3.c_ab_ai_ag_m $24$ (not in LMFDB) 5.3.c_ab_ai_ad_g $24$ (not in LMFDB) 5.3.c_b_ae_m_bk $24$ (not in LMFDB) 5.3.c_d_i_s_bk $24$ (not in LMFDB) 5.3.c_f_e_y_bk $24$ (not in LMFDB) 5.3.c_h_i_s_m $24$ (not in LMFDB) 5.3.c_l_q_cc_ci $24$ (not in LMFDB) 5.3.d_g_ab_av_aci $24$ (not in LMFDB) 5.3.d_g_t_bn_ci $24$ (not in LMFDB) 5.3.d_j_i_m_ag $24$ (not in LMFDB) 5.3.e_c_ak_ap_am $24$ (not in LMFDB) 5.3.e_f_c_j_be $24$ (not in LMFDB) 5.3.e_i_o_bh_cu $24$ (not in LMFDB) 5.3.e_l_ba_cf_ek $24$ (not in LMFDB) 5.3.e_o_bm_dd_ga $24$ (not in LMFDB) 5.3.f_n_u_y_be $24$ (not in LMFDB) 5.3.f_r_bo_dg_fu $24$ (not in LMFDB) 5.3.g_p_i_acc_aga $24$ (not in LMFDB) 5.3.g_p_i_abz_afu $24$ (not in LMFDB) 5.3.g_p_bc_cr_fu $24$ (not in LMFDB) 5.3.g_t_bo_co_ee $24$ (not in LMFDB) 5.3.g_v_bs_cu_ee $24$ (not in LMFDB) 5.3.g_bb_dc_hq_oi $24$ (not in LMFDB) 5.3.h_ba_ct_gd_lo $24$ (not in LMFDB) 5.3.j_bt_fw_ou_bcw $24$ (not in LMFDB) 5.3.k_bv_fk_lx_vy $24$ (not in LMFDB)