# Properties

 Label 5.3.al_cj_aio_wn_absr 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: $( 1 - x + 3 x^{2} )^{2}( 1 - 3 x + 3 x^{2} )^{3}$ Frobenius angles: $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.406785250661$, $\pm0.406785250661$ Angle rank: $1$ (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 $A(\F_{q^r})$ 9 77175 28449792 4238836875 902957021559 249811933593600 60247323228432069 13061567448051451875 2914370282453391369216 697768996307719253754375

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -7 11 44 95 263 872 2597 7031 19412 57371

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ad 3 $\times$ 1.3.ab 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 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-3})$$$)$ 1.3.ab 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-11})$$$)$
Endomorphism algebra over $\overline{\F}_{3}$
 The base change of $A$ to $\F_{3^{6}}$ is 1.729.ak 2 $\times$ 1.729.cc 3 . The endomorphism algebra for each factor is: 1.729.ak 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-11})$$$)$ 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.f 2 . The endomorphism algebra for each factor is: 1.9.ad 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-3})$$$)$ 1.9.f 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-11})$$$)$
• Endomorphism algebra over $\F_{3^{3}}$  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a 3 $\times$ 1.27.i 2 . The endomorphism algebra for each factor is: 1.27.a 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-3})$$$)$ 1.27.i 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-11})$$$)$

## 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_bp_aew_ll_avv $2$ (not in LMFDB) 5.3.ah_z_aco_fx_all $2$ (not in LMFDB) 5.3.af_n_as_j_j $2$ (not in LMFDB) 5.3.ad_f_ag_j_aj $2$ (not in LMFDB) 5.3.ab_b_ag_j_j $2$ (not in LMFDB) 5.3.b_b_g_j_aj $2$ (not in LMFDB) 5.3.d_f_g_j_j $2$ (not in LMFDB) 5.3.f_n_s_j_aj $2$ (not in LMFDB) 5.3.h_z_co_fx_ll $2$ (not in LMFDB) 5.3.j_bp_ew_ll_vv $2$ (not in LMFDB) 5.3.l_cj_io_wn_bsr $2$ (not in LMFDB) 5.3.ai_z_ay_acl_ii $3$ (not in LMFDB) 5.3.ai_bl_aeq_ll_awe $3$ (not in LMFDB) 5.3.af_k_ad_abb_cu $3$ (not in LMFDB) 5.3.af_w_acl_fx_alc $3$ (not in LMFDB) 5.3.ac_af_s_j_acu $3$ (not in LMFDB) 5.3.ac_e_a_aj_bk $3$ (not in LMFDB) 5.3.ac_h_ag_j_a $3$ (not in LMFDB) 5.3.ac_q_ay_dv_aee $3$ (not in LMFDB) 5.3.b_ac_d_j_a $3$ (not in LMFDB) 5.3.b_h_m_s_cc $3$ (not in LMFDB) 5.3.b_k_p_bt_cu $3$ (not in LMFDB) 5.3.e_b_am_j_cu $3$ (not in LMFDB) 5.3.e_k_y_bt_cu $3$ (not in LMFDB) 5.3.e_n_bk_dd_fo $3$ (not in LMFDB) 5.3.h_w_bt_dd_fo $3$ (not in LMFDB) 5.3.k_br_dy_fx_ii $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 5.3.aj_bp_aew_ll_avv $2$ (not in LMFDB) 5.3.ah_z_aco_fx_all $2$ (not in LMFDB) 5.3.af_n_as_j_j $2$ (not in LMFDB) 5.3.ad_f_ag_j_aj $2$ (not in LMFDB) 5.3.ab_b_ag_j_j $2$ (not in LMFDB) 5.3.b_b_g_j_aj $2$ (not in LMFDB) 5.3.d_f_g_j_j $2$ (not in LMFDB) 5.3.f_n_s_j_aj $2$ (not in LMFDB) 5.3.h_z_co_fx_ll $2$ (not in LMFDB) 5.3.j_bp_ew_ll_vv $2$ (not in LMFDB) 5.3.l_cj_io_wn_bsr $2$ (not in LMFDB) 5.3.ai_z_ay_acl_ii $3$ (not in LMFDB) 5.3.ai_bl_aeq_ll_awe $3$ (not in LMFDB) 5.3.af_k_ad_abb_cu $3$ (not in LMFDB) 5.3.af_w_acl_fx_alc $3$ (not in LMFDB) 5.3.ac_af_s_j_acu $3$ (not in LMFDB) 5.3.ac_e_a_aj_bk $3$ (not in LMFDB) 5.3.ac_h_ag_j_a $3$ (not in LMFDB) 5.3.ac_q_ay_dv_aee $3$ (not in LMFDB) 5.3.b_ac_d_j_a $3$ (not in LMFDB) 5.3.b_h_m_s_cc $3$ (not in LMFDB) 5.3.b_k_p_bt_cu $3$ (not in LMFDB) 5.3.e_b_am_j_cu $3$ (not in LMFDB) 5.3.e_k_y_bt_cu $3$ (not in LMFDB) 5.3.e_n_bk_dd_fo $3$ (not in LMFDB) 5.3.h_w_bt_dd_fo $3$ (not in LMFDB) 5.3.k_br_dy_fx_ii $3$ (not in LMFDB) 5.3.aj_bf_abk_acl_jj $4$ (not in LMFDB) 5.3.af_t_abw_eb_ahh $4$ (not in LMFDB) 5.3.ad_af_y_j_adv $4$ (not in LMFDB) 5.3.ad_b_g_ad_aj $4$ (not in LMFDB) 5.3.ad_l_ay_cf_adv $4$ (not in LMFDB) 5.3.ab_h_am_bh_abt $4$ (not in LMFDB) 5.3.b_h_m_bh_bt $4$ (not in LMFDB) 5.3.d_af_ay_j_dv $4$ (not in LMFDB) 5.3.d_b_ag_ad_j $4$ (not in LMFDB) 5.3.d_l_y_cf_dv $4$ (not in LMFDB) 5.3.f_t_bw_eb_hh $4$ (not in LMFDB) 5.3.j_bf_bk_acl_ajj $4$ (not in LMFDB) 5.3.ak_br_ady_fx_aii $6$ (not in LMFDB) 5.3.ah_w_abt_dd_afo $6$ (not in LMFDB) 5.3.ag_x_aco_fx_alc $6$ (not in LMFDB) 5.3.ae_b_m_j_acu $6$ (not in LMFDB) 5.3.ae_k_ay_bt_acu $6$ (not in LMFDB) 5.3.ae_n_abk_dd_afo $6$ (not in LMFDB) 5.3.ad_o_abh_dd_afo $6$ (not in LMFDB) 5.3.ab_ac_ad_j_a $6$ (not in LMFDB) 5.3.ab_h_am_s_acc $6$ (not in LMFDB) 5.3.ab_k_ap_bt_acu $6$ (not in LMFDB) 5.3.a_f_a_j_a $6$ (not in LMFDB) 5.3.a_o_a_dd_a $6$ (not in LMFDB) 5.3.c_af_as_j_cu $6$ (not in LMFDB) 5.3.c_e_a_aj_abk $6$ (not in LMFDB) 5.3.c_h_g_j_a $6$ (not in LMFDB) 5.3.c_q_y_dv_ee $6$ (not in LMFDB) 5.3.d_o_bh_dd_fo $6$ (not in LMFDB) 5.3.f_k_d_abb_acu $6$ (not in LMFDB) 5.3.f_w_cl_fx_lc $6$ (not in LMFDB) 5.3.g_x_co_fx_lc $6$ (not in LMFDB) 5.3.i_z_y_acl_aii $6$ (not in LMFDB) 5.3.i_bl_eq_ll_we $6$ (not in LMFDB) 5.3.ac_h_ap_bb_acl $9$ (not in LMFDB) 5.3.ac_h_d_aj_cl $9$ (not in LMFDB) 5.3.b_ac_ag_a_s $9$ (not in LMFDB) 5.3.b_ac_m_s_as $9$ (not in LMFDB) 5.3.ag_n_ag_abb_cu $12$ (not in LMFDB) 5.3.af_k_ad_abn_ee $12$ (not in LMFDB) 5.3.ae_ac_y_ad_acu $12$ (not in LMFDB) 5.3.ae_h_am_bh_acu $12$ (not in LMFDB) 5.3.ad_ai_bh_p_afo $12$ (not in LMFDB) 5.3.ad_c_d_ap_bk $12$ (not in LMFDB) 5.3.ad_e_ad_aj_bk $12$ (not in LMFDB) 5.3.ac_ai_y_p_aee $12$ (not in LMFDB) 5.3.ac_b_g_ad_a $12$ (not in LMFDB) 5.3.ac_e_a_av_bk $12$ (not in LMFDB) 5.3.ac_n_as_cr_acu $12$ (not in LMFDB) 5.3.ab_af_a_g_s $12$ (not in LMFDB) 5.3.ab_ac_ad_ad_bk $12$ (not in LMFDB) 5.3.ab_e_aj_p_abk $12$ (not in LMFDB) 5.3.a_ai_a_p_a $12$ (not in LMFDB) 5.3.a_af_a_j_a $12$ (not in LMFDB) 5.3.a_b_a_ad_a $12$ (not in LMFDB) 5.3.a_c_a_ap_a $12$ (not in LMFDB) 5.3.a_e_a_aj_a $12$ (not in LMFDB) 5.3.a_l_a_cf_a $12$ (not in LMFDB) 5.3.b_af_a_g_as $12$ (not in LMFDB) 5.3.b_ac_d_ad_abk $12$ (not in LMFDB) 5.3.b_e_j_p_bk $12$ (not in LMFDB) 5.3.c_ai_ay_p_ee $12$ (not in LMFDB) 5.3.c_b_ag_ad_a $12$ (not in LMFDB) 5.3.c_e_a_av_abk $12$ (not in LMFDB) 5.3.c_n_s_cr_cu $12$ (not in LMFDB) 5.3.d_ai_abh_p_fo $12$ (not in LMFDB) 5.3.d_c_ad_ap_abk $12$ (not in LMFDB) 5.3.d_e_d_aj_abk $12$ (not in LMFDB) 5.3.e_ac_ay_ad_cu $12$ (not in LMFDB) 5.3.e_h_m_bh_cu $12$ (not in LMFDB) 5.3.f_k_d_abn_aee $12$ (not in LMFDB) 5.3.g_n_g_abb_acu $12$ (not in LMFDB) 5.3.ab_ac_am_s_s $18$ (not in LMFDB) 5.3.ab_ac_g_a_as $18$ (not in LMFDB) 5.3.a_f_aj_j_abt $18$ (not in LMFDB) 5.3.a_f_j_j_bt $18$ (not in LMFDB) 5.3.c_h_ad_aj_acl $18$ (not in LMFDB) 5.3.c_h_p_bb_cl $18$ (not in LMFDB) 5.3.af_q_abh_cf_adm $24$ (not in LMFDB) 5.3.ae_e_a_v_acu $24$ (not in LMFDB) 5.3.ad_ac_p_d_acc $24$ (not in LMFDB) 5.3.ad_i_ap_bh_acc $24$ (not in LMFDB) 5.3.ac_ac_m_d_abk $24$ (not in LMFDB) 5.3.ac_k_am_bn_abk $24$ (not in LMFDB) 5.3.ab_b_ag_m_as $24$ (not in LMFDB) 5.3.ab_e_aj_v_as $24$ (not in LMFDB) 5.3.a_ac_a_d_a $24$ (not in LMFDB) 5.3.a_i_a_bh_a $24$ (not in LMFDB) 5.3.b_b_g_m_s $24$ (not in LMFDB) 5.3.b_e_j_v_s $24$ (not in LMFDB) 5.3.c_ac_am_d_bk $24$ (not in LMFDB) 5.3.c_k_m_bn_bk $24$ (not in LMFDB) 5.3.d_ac_ap_d_cc $24$ (not in LMFDB) 5.3.d_i_p_bh_cc $24$ (not in LMFDB) 5.3.e_e_a_v_cu $24$ (not in LMFDB) 5.3.f_q_bh_cf_dm $24$ (not in LMFDB) 5.3.a_af_aj_j_bt $36$ (not in LMFDB) 5.3.a_af_j_j_abt $36$ (not in LMFDB)