# Properties

 Label 5.3.ak_bx_aga_of_abba 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: $( 1 - 2 x + 3 x^{2} )( 1 + x + 3 x^{2} )( 1 - 3 x + 3 x^{2} )^{3}$ Frobenius angles: $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.304086723985$, $\pm0.593214749339$ Angle rank: $2$ (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 $A(\F_{q^r})$ 10 61740 16683520 5425711200 1324512107050 237321336913920 51687829984303390 12773919445824700800 3004201127104304350720 709169165824859455646700

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -6 8 30 116 354 836 2262 6884 20010 58328

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ad 3 $\times$ 1.3.ac $\times$ 1.3.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 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-3})$$$)$ 1.3.ac : $$\Q(\sqrt{-2})$$. 1.3.b : $$\Q(\sqrt{-11})$$.
Endomorphism algebra over $\overline{\F}_{3}$
 The base change of $A$ to $\F_{3^{6}}$ is 1.729.abu $\times$ 1.729.ak $\times$ 1.729.cc 3 . The endomorphism algebra for each factor is: 1.729.abu : $$\Q(\sqrt{-2})$$. 1.729.ak : $$\Q(\sqrt{-11})$$. 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 $\times$ 1.9.f. The endomorphism algebra for each factor is: 1.9.ad 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-3})$$$)$ 1.9.c : $$\Q(\sqrt{-2})$$. 1.9.f : $$\Q(\sqrt{-11})$$.
• Endomorphism algebra over $\F_{3^{3}}$  The base change of $A$ to $\F_{3^{3}}$ is 1.27.ai $\times$ 1.27.a 3 $\times$ 1.27.k. The endomorphism algebra for each factor is: 1.27.ai : $$\Q(\sqrt{-11})$$. 1.27.a 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-3})$$$)$ 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.am_ct_akk_bcb_acec $2$ (not in LMFDB) 5.3.ai_bf_ada_fx_akk $2$ (not in LMFDB) 5.3.ag_r_abk_dd_agg $2$ (not in LMFDB) 5.3.ag_r_ay_j_s $2$ (not in LMFDB) 5.3.ae_h_ag_j_as $2$ (not in LMFDB) 5.3.ac_b_a_j_as $2$ (not in LMFDB) 5.3.a_ab_ag_j_s $2$ (not in LMFDB) 5.3.a_ab_g_j_as $2$ (not in LMFDB) 5.3.c_b_a_j_s $2$ (not in LMFDB) 5.3.e_h_g_j_s $2$ (not in LMFDB) 5.3.g_r_y_j_as $2$ (not in LMFDB) 5.3.g_r_bk_dd_gg $2$ (not in LMFDB) 5.3.i_bf_da_fx_kk $2$ (not in LMFDB) 5.3.k_bx_ga_of_bba $2$ (not in LMFDB) 5.3.m_ct_kk_bcb_cec $2$ (not in LMFDB) 5.3.ah_bc_add_hh_anw $3$ (not in LMFDB) 5.3.ae_h_ag_j_as $3$ (not in LMFDB) 5.3.ae_q_abq_dv_agy $3$ (not in LMFDB) 5.3.ab_e_ad_j_a $3$ (not in LMFDB) 5.3.ab_n_am_cu_acc $3$ (not in LMFDB) 5.3.c_b_a_j_s $3$ (not in LMFDB) 5.3.c_k_s_bt_cu $3$ (not in LMFDB) 5.3.f_q_bn_dd_fo $3$ (not in LMFDB) 5.3.i_bf_da_fx_kk $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 5.3.am_ct_akk_bcb_acec $2$ (not in LMFDB) 5.3.ai_bf_ada_fx_akk $2$ (not in LMFDB) 5.3.ag_r_abk_dd_agg $2$ (not in LMFDB) 5.3.ag_r_ay_j_s $2$ (not in LMFDB) 5.3.ae_h_ag_j_as $2$ (not in LMFDB) 5.3.ac_b_a_j_as $2$ (not in LMFDB) 5.3.a_ab_ag_j_s $2$ (not in LMFDB) 5.3.a_ab_g_j_as $2$ (not in LMFDB) 5.3.c_b_a_j_s $2$ (not in LMFDB) 5.3.e_h_g_j_s $2$ (not in LMFDB) 5.3.g_r_y_j_as $2$ (not in LMFDB) 5.3.g_r_bk_dd_gg $2$ (not in LMFDB) 5.3.i_bf_da_fx_kk $2$ (not in LMFDB) 5.3.k_bx_ga_of_bba $2$ (not in LMFDB) 5.3.m_ct_kk_bcb_cec $2$ (not in LMFDB) 5.3.ah_bc_add_hh_anw $3$ (not in LMFDB) 5.3.ae_h_ag_j_as $3$ (not in LMFDB) 5.3.ae_q_abq_dv_agy $3$ (not in LMFDB) 5.3.ab_e_ad_j_a $3$ (not in LMFDB) 5.3.ab_n_am_cu_acc $3$ (not in LMFDB) 5.3.c_b_a_j_s $3$ (not in LMFDB) 5.3.c_k_s_bt_cu $3$ (not in LMFDB) 5.3.f_q_bn_dd_fo $3$ (not in LMFDB) 5.3.i_bf_da_fx_kk $3$ (not in LMFDB) 5.3.ag_x_aci_ez_aja $4$ (not in LMFDB) 5.3.ae_n_abe_cr_aew $4$ (not in LMFDB) 5.3.ac_h_am_bh_acc $4$ (not in LMFDB) 5.3.a_f_ag_v_as $4$ (not in LMFDB) 5.3.a_f_g_v_s $4$ (not in LMFDB) 5.3.c_h_m_bh_cc $4$ (not in LMFDB) 5.3.e_n_be_cr_ew $4$ (not in LMFDB) 5.3.g_x_ci_ez_ja $4$ (not in LMFDB) 5.3.aj_bs_afr_of_abbs $6$ (not in LMFDB) 5.3.ag_ba_ada_hh_anw $6$ (not in LMFDB) 5.3.af_q_abn_dd_afo $6$ (not in LMFDB) 5.3.ad_i_av_bt_acu $6$ (not in LMFDB) 5.3.ad_i_aj_j_a $6$ (not in LMFDB) 5.3.ad_r_abk_ee_agg $6$ (not in LMFDB) 5.3.ac_k_as_bt_acu $6$ (not in LMFDB) 5.3.a_i_ag_bb_abk $6$ (not in LMFDB) 5.3.a_i_g_bb_bk $6$ (not in LMFDB) 5.3.b_e_d_j_a $6$ (not in LMFDB) 5.3.b_n_m_cu_cc $6$ (not in LMFDB) 5.3.d_i_j_j_a $6$ (not in LMFDB) 5.3.d_i_v_bt_cu $6$ (not in LMFDB) 5.3.d_r_bk_ee_gg $6$ (not in LMFDB) 5.3.e_q_bq_dv_gy $6$ (not in LMFDB) 5.3.g_ba_da_hh_nw $6$ (not in LMFDB) 5.3.h_bc_dd_hh_nw $6$ (not in LMFDB) 5.3.j_bs_fr_of_bbs $6$ (not in LMFDB) 5.3.ab_e_am_s_abk $9$ (not in LMFDB) 5.3.ab_e_g_a_bk $9$ (not in LMFDB) 5.3.ag_o_ag_abz_fo $12$ (not in LMFDB) 5.3.ae_e_g_av_bk $12$ (not in LMFDB) 5.3.ad_f_a_ay_cc $12$ (not in LMFDB) 5.3.ad_o_abb_cx_aee $12$ (not in LMFDB) 5.3.ac_ac_g_ad_a $12$ (not in LMFDB) 5.3.ab_b_a_am_s $12$ (not in LMFDB) 5.3.ab_k_aj_bz_abk $12$ (not in LMFDB) 5.3.a_ae_ag_d_bk $12$ (not in LMFDB) 5.3.a_ae_g_d_abk $12$ (not in LMFDB) 5.3.b_b_a_am_as $12$ (not in LMFDB) 5.3.b_k_j_bz_bk $12$ (not in LMFDB) 5.3.c_ac_ag_ad_a $12$ (not in LMFDB) 5.3.d_f_a_ay_acc $12$ (not in LMFDB) 5.3.d_o_bb_cx_ee $12$ (not in LMFDB) 5.3.e_e_ag_av_abk $12$ (not in LMFDB) 5.3.g_o_g_abz_afo $12$ (not in LMFDB) 5.3.ad_i_as_bk_acu $18$ (not in LMFDB) 5.3.ad_i_a_as_cu $18$ (not in LMFDB) 5.3.b_e_ag_a_abk $18$ (not in LMFDB) 5.3.b_e_m_s_bk $18$ (not in LMFDB) 5.3.d_i_a_as_acu $18$ (not in LMFDB) 5.3.d_i_s_bk_cu $18$ (not in LMFDB) 5.3.ag_u_abq_cr_aee $24$ (not in LMFDB) 5.3.ae_k_as_bn_acu $24$ (not in LMFDB) 5.3.ad_l_as_bq_acc $24$ (not in LMFDB) 5.3.ac_e_ag_v_abk $24$ (not in LMFDB) 5.3.ab_h_ag_be_as $24$ (not in LMFDB) 5.3.a_c_ag_p_a $24$ (not in LMFDB) 5.3.a_c_g_p_a $24$ (not in LMFDB) 5.3.b_h_g_be_s $24$ (not in LMFDB) 5.3.c_e_g_v_bk $24$ (not in LMFDB) 5.3.d_l_s_bq_cc $24$ (not in LMFDB) 5.3.e_k_s_bn_cu $24$ (not in LMFDB) 5.3.g_u_bq_cr_ee $24$ (not in LMFDB)