# Properties

 Label 5.3.ak_bv_afk_lx_avy 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: $( 1 - 2 x + 3 x^{2} )( 1 - 3 x + 3 x^{2} )^{2}( 1 - 2 x + x^{2} - 6 x^{3} + 9 x^{4} )$ Frobenius angles: $\pm0.0292466093486$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.304086723985$, $\pm0.637420057318$ 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 $A(\F_{q^r})$ 6 33516 9652608 4395422304 1036558465206 196698220314624 49780651221603042 12463487624210313600 2926266810943148717184 712025827391529486333036

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -6 4 18 100 294 700 2178 6724 19494 58564

## 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.ac_b 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.ac_b : $$\Q(\sqrt{-2}, \sqrt{-3})$$.
Endomorphism algebra over $\overline{\F}_{3}$
 The base change of $A$ to $\F_{3^{6}}$ is 1.729.abu 3 $\times$ 1.729.cc 2 . The endomorphism algebra for each factor is: 1.729.abu 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-2})$$$)$ 1.729.cc 2 : $\mathrm{M}_{2}(B)$, 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 2 $\times$ 1.9.c $\times$ 2.9.ac_af. 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.ac_af : $$\Q(\sqrt{-2}, \sqrt{-3})$$.
• Endomorphism algebra over $\F_{3^{3}}$  The base change of $A$ to $\F_{3^{3}}$ is 1.27.ak 2 $\times$ 1.27.a 2 $\times$ 1.27.k. The endomorphism algebra for each factor is: 1.27.ak 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-2})$$$)$ 1.27.a 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$ 1.27.k : $$\Q(\sqrt{-2})$$.

## 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.ag_p_abc_cr_afu $2$ (not in LMFDB) 5.3.ag_p_ai_abz_fu $2$ (not in LMFDB) 5.3.ae_f_ac_j_abe $2$ (not in LMFDB) 5.3.ac_ab_i_ad_ag $2$ (not in LMFDB) 5.3.a_ad_ak_j_be $2$ (not in LMFDB) 5.3.a_ad_k_j_abe $2$ (not in LMFDB) 5.3.c_ab_ai_ad_g $2$ (not in LMFDB) 5.3.e_f_c_j_be $2$ (not in LMFDB) 5.3.g_p_i_abz_afu $2$ (not in LMFDB) 5.3.g_p_bc_cr_fu $2$ (not in LMFDB) 5.3.k_bv_fk_lx_vy $2$ (not in LMFDB) 5.3.ah_ba_act_gd_alo $3$ (not in LMFDB) 5.3.ae_f_ac_j_abe $3$ (not in LMFDB) 5.3.ae_i_ao_bn_adg $3$ (not in LMFDB) 5.3.ae_o_abm_dd_aga $3$ (not in LMFDB) 5.3.ab_c_af_d_am $3$ (not in LMFDB) 5.3.ab_f_ai_y_abe $3$ (not in LMFDB) 5.3.c_ab_ai_ad_g $3$ (not in LMFDB) 5.3.c_c_ac_j_y $3$ (not in LMFDB) 5.3.c_l_q_cc_ci $3$ (not in LMFDB) 5.3.f_r_bo_dg_fu $3$ (not in LMFDB) 5.3.i_bg_de_gd_kq $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 5.3.ag_p_abc_cr_afu $2$ (not in LMFDB) 5.3.ag_p_ai_abz_fu $2$ (not in LMFDB) 5.3.ae_f_ac_j_abe $2$ (not in LMFDB) 5.3.ac_ab_i_ad_ag $2$ (not in LMFDB) 5.3.a_ad_ak_j_be $2$ (not in LMFDB) 5.3.a_ad_k_j_abe $2$ (not in LMFDB) 5.3.c_ab_ai_ad_g $2$ (not in LMFDB) 5.3.e_f_c_j_be $2$ (not in LMFDB) 5.3.g_p_i_abz_afu $2$ (not in LMFDB) 5.3.g_p_bc_cr_fu $2$ (not in LMFDB) 5.3.k_bv_fk_lx_vy $2$ (not in LMFDB) 5.3.ah_ba_act_gd_alo $3$ (not in LMFDB) 5.3.ae_f_ac_j_abe $3$ (not in LMFDB) 5.3.ae_i_ao_bn_adg $3$ (not in LMFDB) 5.3.ae_o_abm_dd_aga $3$ (not in LMFDB) 5.3.ab_c_af_d_am $3$ (not in LMFDB) 5.3.ab_f_ai_y_abe $3$ (not in LMFDB) 5.3.c_ab_ai_ad_g $3$ (not in LMFDB) 5.3.c_c_ac_j_y $3$ (not in LMFDB) 5.3.c_l_q_cc_ci $3$ (not in LMFDB) 5.3.f_r_bo_dg_fu $3$ (not in LMFDB) 5.3.i_bg_de_gd_kq $3$ (not in LMFDB) 5.3.ae_l_aba_cf_aek $4$ (not in LMFDB) 5.3.a_d_ak_j_abe $4$ (not in LMFDB) 5.3.a_d_k_j_be $4$ (not in LMFDB) 5.3.e_l_ba_cf_ek $4$ (not in LMFDB) 5.3.am_cu_aks_bdf_acgq $6$ (not in LMFDB) 5.3.aj_bt_afw_ou_abcw $6$ (not in LMFDB) 5.3.ai_bg_ade_gd_akq $6$ (not in LMFDB) 5.3.ag_s_aba_j_y $6$ (not in LMFDB) 5.3.ag_bb_adc_hq_aoi $6$ (not in LMFDB) 5.3.af_r_abo_dg_afu $6$ (not in LMFDB) 5.3.ad_g_at_bn_aci $6$ (not in LMFDB) 5.3.ad_g_b_av_ci $6$ (not in LMFDB) 5.3.ad_j_ai_m_g $6$ (not in LMFDB) 5.3.ac_c_c_j_ay $6$ (not in LMFDB) 5.3.ac_l_aq_cc_aci $6$ (not in LMFDB) 5.3.a_a_ak_p_m $6$ (not in LMFDB) 5.3.a_a_k_p_am $6$ (not in LMFDB) 5.3.a_g_ak_j_aci $6$ (not in LMFDB) 5.3.a_g_k_j_ci $6$ (not in LMFDB) 5.3.b_c_f_d_m $6$ (not in LMFDB) 5.3.b_f_i_y_be $6$ (not in LMFDB) 5.3.d_g_ab_av_aci $6$ (not in LMFDB) 5.3.d_g_t_bn_ci $6$ (not in LMFDB) 5.3.d_j_i_m_ag $6$ (not in LMFDB) 5.3.e_i_o_bn_dg $6$ (not in LMFDB) 5.3.e_o_bm_dd_ga $6$ (not in LMFDB) 5.3.g_s_ba_j_ay $6$ (not in LMFDB) 5.3.g_bb_dc_hq_oi $6$ (not in LMFDB) 5.3.h_ba_ct_gd_lo $6$ (not in LMFDB) 5.3.j_bt_fw_ou_bcw $6$ (not in LMFDB) 5.3.m_cu_ks_bdf_cgq $6$ (not in LMFDB) 5.3.ai_bc_aby_bn_am $12$ (not in LMFDB) 5.3.ag_p_ai_acc_ga $12$ (not in LMFDB) 5.3.ag_y_ack_ff_ajg $12$ (not in LMFDB) 5.3.af_n_au_y_abe $12$ (not in LMFDB) 5.3.ae_c_k_ap_m $12$ (not in LMFDB) 5.3.ae_e_c_p_aci $12$ (not in LMFDB) 5.3.ac_af_q_g_aci $12$ (not in LMFDB) 5.3.ac_ac_k_j_abw $12$ (not in LMFDB) 5.3.ac_ab_i_ag_am $12$ (not in LMFDB) 5.3.ac_e_ac_p_ay $12$ (not in LMFDB) 5.3.ac_h_ai_s_am $12$ (not in LMFDB) 5.3.ac_i_ak_bn_abw $12$ (not in LMFDB) 5.3.ab_b_ae_m_ag $12$ (not in LMFDB) 5.3.a_ag_ak_j_ci $12$ (not in LMFDB) 5.3.a_ag_k_j_aci $12$ (not in LMFDB) 5.3.b_b_e_m_g $12$ (not in LMFDB) 5.3.c_af_aq_g_ci $12$ (not in LMFDB) 5.3.c_ac_ak_j_bw $12$ (not in LMFDB) 5.3.c_ab_ai_ag_m $12$ (not in LMFDB) 5.3.c_e_c_p_y $12$ (not in LMFDB) 5.3.c_h_i_s_m $12$ (not in LMFDB) 5.3.c_i_k_bn_bw $12$ (not in LMFDB) 5.3.e_c_ak_ap_am $12$ (not in LMFDB) 5.3.e_e_ac_p_ci $12$ (not in LMFDB) 5.3.f_n_u_y_be $12$ (not in LMFDB) 5.3.g_p_i_acc_aga $12$ (not in LMFDB) 5.3.g_y_ck_ff_jg $12$ (not in LMFDB) 5.3.i_bc_by_bn_m $12$ (not in LMFDB) 5.3.am_cs_akc_bax_acbo $24$ (not in LMFDB) 5.3.aj_br_afm_nq_abao $24$ (not in LMFDB) 5.3.ai_be_acw_fr_ake $24$ (not in LMFDB) 5.3.ag_n_ae_abw_fc $24$ (not in LMFDB) 5.3.ag_q_aw_j_m $24$ (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_w_acg_et_aiu $24$ (not in LMFDB) 5.3.ag_z_acy_gy_ank $24$ (not in LMFDB) 5.3.af_p_abm_da_afi $24$ (not in LMFDB) 5.3.ae_g_c_av_bw $24$ (not in LMFDB) 5.3.ae_i_ao_bh_acu $24$ (not in LMFDB) 5.3.ad_h_ak_g_ag $24$ (not in LMFDB) 5.3.ac_ad_e_a_m $24$ (not in LMFDB) 5.3.ac_a_ac_j_am $24$ (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_g_ao_bb_aci $24$ (not in LMFDB) 5.3.ac_j_au_bk_adg $24$ (not in LMFDB) 5.3.ab_d_c_ag_be $24$ (not in LMFDB) 5.3.a_ac_ac_d_y $24$ (not in LMFDB) 5.3.a_ac_c_d_ay $24$ (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.b_d_ac_ag_abe $24$ (not in LMFDB) 5.3.c_ad_ae_a_am $24$ (not in LMFDB) 5.3.c_a_c_j_m $24$ (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_g_o_bb_ci $24$ (not in LMFDB) 5.3.c_j_u_bk_dg $24$ (not in LMFDB) 5.3.d_h_k_g_g $24$ (not in LMFDB) 5.3.e_g_ac_av_abw $24$ (not in LMFDB) 5.3.e_i_o_bh_cu $24$ (not in LMFDB) 5.3.f_p_bm_da_fi $24$ (not in LMFDB) 5.3.g_n_e_abw_afc $24$ (not in LMFDB) 5.3.g_q_w_j_am $24$ (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_w_cg_et_iu $24$ (not in LMFDB) 5.3.g_z_cy_gy_nk $24$ (not in LMFDB) 5.3.i_be_cw_fr_ke $24$ (not in LMFDB) 5.3.j_br_fm_nq_bao $24$ (not in LMFDB) 5.3.m_cs_kc_bax_cbo $24$ (not in LMFDB)