# Properties

 Label 5.3.al_ch_aia_us_abow Base Field $\F_{3}$ Dimension $5$ Ordinary No $p$-rank $4$ Principally polarizable Yes Contains a Jacobian No

## Invariants

 Base field: $\F_{3}$ Dimension: $5$ L-polynomial: $( 1 - 3 x + 3 x^{2} )( 1 - 4 x + 8 x^{2} - 12 x^{3} + 9 x^{4} )^{2}$ Frobenius angles: $\pm0.0540867239847$, $\pm0.0540867239847$, $\pm0.166666666667$, $\pm0.445913276015$, $\pm0.445913276015$ 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 $A(\F_{q^r})$ 4 32368 10972528 1945705216 658181907004 221974241440000 54436404955943476 12063819633480695808 2868089541454266945424 715002677300852642155888

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -7 7 20 35 183 784 2373 6507 19100 58807

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ad $\times$ 2.3.ae_i 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 : $$\Q(\sqrt{-3})$$. 2.3.ae_i 2 : $\mathrm{M}_{2}($$$\Q(\zeta_{8})$$$)$
Endomorphism algebra over $\overline{\F}_{3}$
 The base change of $A$ to $\F_{3^{12}}$ is 1.531441.acec $\times$ 1.531441.zi 4 . The endomorphism algebra for each factor is: 1.531441.acec : the quaternion algebra over $$\Q$$ ramified at $3$ and $\infty$. 1.531441.zi 4 : $\mathrm{M}_{4}($$$\Q(\sqrt{-2})$$$)$
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 $\times$ 2.9.a_ao 2 . The endomorphism algebra for each factor is: 1.9.ad : $$\Q(\sqrt{-3})$$. 2.9.a_ao 2 : $\mathrm{M}_{2}($$$\Q(\zeta_{8})$$$)$
• Endomorphism algebra over $\F_{3^{3}}$  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a $\times$ 2.27.ae_i 2 . The endomorphism algebra for each factor is: 1.27.a : $$\Q(\sqrt{-3})$$. 2.27.ae_i 2 : $\mathrm{M}_{2}($$$\Q(\zeta_{8})$$$)$
• Endomorphism algebra over $\F_{3^{4}}$  The base change of $A$ to $\F_{3^{4}}$ is 1.81.ao 4 $\times$ 1.81.j. The endomorphism algebra for each factor is: 1.81.ao 4 : $\mathrm{M}_{4}($$$\Q(\sqrt{-2})$$$)$ 1.81.j : $$\Q(\sqrt{-3})$$.
• Endomorphism algebra over $\F_{3^{6}}$  The base change of $A$ to $\F_{3^{6}}$ is 1.729.cc $\times$ 2.729.a_zi 2 . The endomorphism algebra for each factor is: 1.729.cc : the quaternion algebra over $$\Q$$ ramified at $3$ and $\infty$. 2.729.a_zi 2 : $\mathrm{M}_{2}($$$\Q(\zeta_{8})$$$)$

## 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.af_l_aq_k_g $2$ (not in LMFDB) 5.3.ad_d_a_ao_bq $2$ (not in LMFDB) 5.3.d_d_a_ao_abq $2$ (not in LMFDB) 5.3.f_l_q_k_ag $2$ (not in LMFDB) 5.3.l_ch_ia_us_bow $2$ (not in LMFDB) 5.3.ai_bj_aei_ko_aui $3$ (not in LMFDB) 5.3.af_l_aq_k_g $3$ (not in LMFDB) 5.3.b_ab_ae_h_bb $3$ (not in LMFDB) 5.3.e_l_u_bf_bw $3$ (not in LMFDB) 5.3.h_x_bs_cd_cr $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 5.3.af_l_aq_k_g $2$ (not in LMFDB) 5.3.ad_d_a_ao_bq $2$ (not in LMFDB) 5.3.d_d_a_ao_abq $2$ (not in LMFDB) 5.3.f_l_q_k_ag $2$ (not in LMFDB) 5.3.l_ch_ia_us_bow $2$ (not in LMFDB) 5.3.ai_bj_aei_ko_aui $3$ (not in LMFDB) 5.3.af_l_aq_k_g $3$ (not in LMFDB) 5.3.b_ab_ae_h_bb $3$ (not in LMFDB) 5.3.e_l_u_bf_bw $3$ (not in LMFDB) 5.3.h_x_bs_cd_cr $3$ (not in LMFDB) 5.3.ah_x_abs_cd_acr $6$ (not in LMFDB) 5.3.ae_l_au_bf_abw $6$ (not in LMFDB) 5.3.ab_ab_e_h_abb $6$ (not in LMFDB) 5.3.a_d_a_ao_a $6$ (not in LMFDB) 5.3.i_bj_ei_ko_ui $6$ (not in LMFDB) 5.3.al_cj_aio_wm_abso $8$ (not in LMFDB) 5.3.al_cl_ajc_yk_abws $8$ (not in LMFDB) 5.3.ah_v_abi_bg_abe $8$ (not in LMFDB) 5.3.ah_x_abo_bi_as $8$ (not in LMFDB) 5.3.ah_z_ack_eu_aio $8$ (not in LMFDB) 5.3.ah_bb_acq_fe_aja $8$ (not in LMFDB) 5.3.af_n_as_i_g $8$ (not in LMFDB) 5.3.af_p_au_k_s $8$ (not in LMFDB) 5.3.ad_ab_m_k_aco $8$ (not in LMFDB) 5.3.ad_d_a_o_abq $8$ (not in LMFDB) 5.3.ad_f_ao_bg_acc $8$ (not in LMFDB) 5.3.ad_f_c_aq_bq $8$ (not in LMFDB) 5.3.ad_h_am_bi_aco $8$ (not in LMFDB) 5.3.ab_ad_c_i_as $8$ (not in LMFDB) 5.3.ab_ab_i_k_abe $8$ (not in LMFDB) 5.3.ab_b_ac_e_as $8$ (not in LMFDB) 5.3.ab_d_e_o_ag $8$ (not in LMFDB) 5.3.b_ad_ac_i_s $8$ (not in LMFDB) 5.3.b_ab_ai_k_be $8$ (not in LMFDB) 5.3.b_b_c_e_s $8$ (not in LMFDB) 5.3.b_d_ae_o_g $8$ (not in LMFDB) 5.3.d_ab_am_k_co $8$ (not in LMFDB) 5.3.d_d_a_o_bq $8$ (not in LMFDB) 5.3.d_f_ac_aq_abq $8$ (not in LMFDB) 5.3.d_f_o_bg_cc $8$ (not in LMFDB) 5.3.d_h_m_bi_co $8$ (not in LMFDB) 5.3.f_n_s_i_ag $8$ (not in LMFDB) 5.3.f_p_u_k_as $8$ (not in LMFDB) 5.3.h_v_bi_bg_be $8$ (not in LMFDB) 5.3.h_x_bo_bi_s $8$ (not in LMFDB) 5.3.h_z_ck_eu_io $8$ (not in LMFDB) 5.3.h_bb_cq_fe_ja $8$ (not in LMFDB) 5.3.l_cj_io_wm_bso $8$ (not in LMFDB) 5.3.l_cl_jc_yk_bws $8$ (not in LMFDB) 5.3.ah_x_abs_cd_acr $12$ (not in LMFDB) 5.3.ad_af_y_i_ads $16$ (not in LMFDB) 5.3.ad_l_ay_ce_ads $16$ (not in LMFDB) 5.3.d_af_ay_i_ds $16$ (not in LMFDB) 5.3.d_l_y_ce_ds $16$ (not in LMFDB) 5.3.aj_bm_aed_jf_ari $24$ (not in LMFDB) 5.3.aj_bo_aen_jz_asm $24$ (not in LMFDB) 5.3.ai_bl_aeq_lk_awe $24$ (not in LMFDB) 5.3.ai_bn_aey_mk_aya $24$ (not in LMFDB) 5.3.ah_v_abu_ef_air $24$ (not in LMFDB) 5.3.ag_u_ace_ev_aiu $24$ (not in LMFDB) 5.3.ag_w_aci_fd_ajs $24$ (not in LMFDB) 5.3.af_i_af_t_aci $24$ (not in LMFDB) 5.3.af_k_ad_abl_dy $24$ (not in LMFDB) 5.3.af_m_az_ch_aeq $24$ (not in LMFDB) 5.3.af_m_af_abp_eq $24$ (not in LMFDB) 5.3.ae_j_abc_cj_ads $24$ (not in LMFDB) 5.3.ae_j_aq_u_ay $24$ (not in LMFDB) 5.3.ae_l_aq_w_ay $24$ (not in LMFDB) 5.3.ae_n_abg_cm_aeq $24$ (not in LMFDB) 5.3.ae_p_abg_cw_aeq $24$ (not in LMFDB) 5.3.ad_b_g_al_p $24$ (not in LMFDB) 5.3.ad_c_af_l_ag $24$ (not in LMFDB) 5.3.ad_e_ad_h_ay $24$ (not in LMFDB) 5.3.ad_f_ag_b_p $24$ (not in LMFDB) 5.3.ac_c_ai_n_am $24$ (not in LMFDB) 5.3.ac_e_a_at_bk $24$ (not in LMFDB) 5.3.ac_g_aq_bd_aci $24$ (not in LMFDB) 5.3.ac_g_e_al_ci $24$ (not in LMFDB) 5.3.ab_ae_l_h_abk $24$ (not in LMFDB) 5.3.ab_ad_ak_n_bh $24$ (not in LMFDB) 5.3.ab_ac_ad_ab_be $24$ (not in LMFDB) 5.3.ab_a_an_t_a $24$ (not in LMFDB) 5.3.ab_a_h_ab_a $24$ (not in LMFDB) 5.3.a_ab_a_k_a $24$ (not in LMFDB) 5.3.a_b_a_al_a $24$ (not in LMFDB) 5.3.a_d_a_o_a $24$ (not in LMFDB) 5.3.a_f_ai_i_abw $24$ (not in LMFDB) 5.3.a_f_a_b_a $24$ (not in LMFDB) 5.3.a_f_i_i_bw $24$ (not in LMFDB) 5.3.a_h_a_bi_a $24$ (not in LMFDB) 5.3.b_ae_al_h_bk $24$ (not in LMFDB) 5.3.b_ad_k_n_abh $24$ (not in LMFDB) 5.3.b_ac_d_ab_abe $24$ (not in LMFDB) 5.3.b_a_ah_ab_a $24$ (not in LMFDB) 5.3.b_a_n_t_a $24$ (not in LMFDB) 5.3.c_c_i_n_m $24$ (not in LMFDB) 5.3.c_e_a_at_abk $24$ (not in LMFDB) 5.3.c_g_ae_al_aci $24$ (not in LMFDB) 5.3.c_g_q_bd_ci $24$ (not in LMFDB) 5.3.d_b_ag_al_ap $24$ (not in LMFDB) 5.3.d_c_f_l_g $24$ (not in LMFDB) 5.3.d_e_d_h_y $24$ (not in LMFDB) 5.3.d_f_g_b_ap $24$ (not in LMFDB) 5.3.e_j_q_u_y $24$ (not in LMFDB) 5.3.e_j_bc_cj_ds $24$ (not in LMFDB) 5.3.e_l_q_w_y $24$ (not in LMFDB) 5.3.e_n_bg_cm_eq $24$ (not in LMFDB) 5.3.e_p_bg_cw_eq $24$ (not in LMFDB) 5.3.f_i_f_t_ci $24$ (not in LMFDB) 5.3.f_k_d_abl_ady $24$ (not in LMFDB) 5.3.f_m_f_abp_aeq $24$ (not in LMFDB) 5.3.f_m_z_ch_eq $24$ (not in LMFDB) 5.3.g_u_ce_ev_iu $24$ (not in LMFDB) 5.3.g_w_ci_fd_js $24$ (not in LMFDB) 5.3.h_v_bu_ef_ir $24$ (not in LMFDB) 5.3.i_bl_eq_lk_we $24$ (not in LMFDB) 5.3.i_bn_ey_mk_ya $24$ (not in LMFDB) 5.3.j_bm_ed_jf_ri $24$ (not in LMFDB) 5.3.j_bo_en_jz_sm $24$ (not in LMFDB) 5.3.af_k_af_au_cf $40$ (not in LMFDB) 5.3.ab_ac_ab_e_j $40$ (not in LMFDB) 5.3.b_ac_b_e_aj $40$ (not in LMFDB) 5.3.f_k_f_au_acf $40$ (not in LMFDB) 5.3.a_af_a_i_a $48$ (not in LMFDB) 5.3.a_l_a_ce_a $48$ (not in LMFDB) 5.3.ac_e_ac_ai_y $120$ (not in LMFDB) 5.3.c_e_c_ai_ay $120$ (not in LMFDB)