# Properties

 Label 5.3.ak_bt_aeq_ir_aoo Base Field $\F_{3}$ Dimension $5$ Ordinary No $p$-rank $1$ Principally polarizable Yes Contains a Jacobian No

## Invariants

 Base field: $\F_{3}$ Dimension: $5$ L-polynomial: $( 1 + 2 x + 3 x^{2} )( 1 - 3 x + 3 x^{2} )^{4}$ Frobenius angles: $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.695913276015$ Angle rank: $1$ (numerical)

This isogeny class is not simple.

## Newton polygon

 $p$-rank: $1$ Slopes: $[0, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1]$

## 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 28812 11063808 6583196256 1326820798326 258416566861824 60273782122893954 12712681180316627328 2926539479384975107584 711977996323730431211532

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -6 0 18 132 354 900 2598 6852 19494 58560

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ad 4 $\times$ 1.3.c 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 4 : $\mathrm{M}_{4}($$$\Q(\sqrt{-3})$$$)$ 1.3.c : $$\Q(\sqrt{-2})$$.
Endomorphism algebra over $\overline{\F}_{3}$
 The base change of $A$ to $\F_{3^{6}}$ is 1.729.abu $\times$ 1.729.cc 4 . The endomorphism algebra for each factor is: 1.729.abu : $$\Q(\sqrt{-2})$$. 1.729.cc 4 : $\mathrm{M}_{4}(B)$, 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 4 $\times$ 1.9.c. The endomorphism algebra for each factor is: 1.9.ad 4 : $\mathrm{M}_{4}($$$\Q(\sqrt{-3})$$$)$ 1.9.c : $$\Q(\sqrt{-2})$$.
• Endomorphism algebra over $\F_{3^{3}}$  The base change of $A$ to $\F_{3^{3}}$ is 1.27.ak $\times$ 1.27.a 4 . The endomorphism algebra for each factor is: 1.27.ak : $$\Q(\sqrt{-2})$$. 1.27.a 4 : $\mathrm{M}_{4}($$$\Q(\sqrt{-3})$$$)$

## 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.ao_dp_aou_bpx_adhe $2$ (not in LMFDB) 5.3.ai_bb_abq_j_cc $2$ (not in LMFDB) 5.3.ac_ad_m_j_acc $2$ (not in LMFDB) 5.3.e_d_ag_j_cc $2$ (not in LMFDB) 5.3.k_bt_eq_ir_oo $2$ (not in LMFDB) 5.3.ah_y_acf_en_aii $3$ (not in LMFDB) 5.3.ae_d_g_j_acc $3$ (not in LMFDB) 5.3.ae_m_abe_cl_aee $3$ (not in LMFDB) 5.3.ab_a_ad_j_a $3$ (not in LMFDB) 5.3.ab_j_am_bk_acc $3$ (not in LMFDB) 5.3.c_ad_am_j_cc $3$ (not in LMFDB) 5.3.c_g_g_j_a $3$ (not in LMFDB) 5.3.c_p_y_dm_ee $3$ (not in LMFDB) 5.3.f_m_p_j_a $3$ (not in LMFDB) 5.3.f_v_ci_fo_kk $3$ (not in LMFDB) 5.3.i_bb_bq_j_acc $3$ (not in LMFDB) 5.3.i_bk_ek_kt_uu $3$ (not in LMFDB) 5.3.l_ci_if_vd_bpo $3$ (not in LMFDB) 5.3.o_dp_ou_bpx_dhe $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 5.3.ao_dp_aou_bpx_adhe $2$ (not in LMFDB) 5.3.ai_bb_abq_j_cc $2$ (not in LMFDB) 5.3.ac_ad_m_j_acc $2$ (not in LMFDB) 5.3.e_d_ag_j_cc $2$ (not in LMFDB) 5.3.k_bt_eq_ir_oo $2$ (not in LMFDB) 5.3.ah_y_acf_en_aii $3$ (not in LMFDB) 5.3.ae_d_g_j_acc $3$ (not in LMFDB) 5.3.ae_m_abe_cl_aee $3$ (not in LMFDB) 5.3.ab_a_ad_j_a $3$ (not in LMFDB) 5.3.ab_j_am_bk_acc $3$ (not in LMFDB) 5.3.c_ad_am_j_cc $3$ (not in LMFDB) 5.3.c_g_g_j_a $3$ (not in LMFDB) 5.3.c_p_y_dm_ee $3$ (not in LMFDB) 5.3.f_m_p_j_a $3$ (not in LMFDB) 5.3.f_v_ci_fo_kk $3$ (not in LMFDB) 5.3.i_bb_bq_j_acc $3$ (not in LMFDB) 5.3.i_bk_ek_kt_uu $3$ (not in LMFDB) 5.3.l_ci_if_vd_bpo $3$ (not in LMFDB) 5.3.o_dp_ou_bpx_dhe $3$ (not in LMFDB) 5.3.ai_bh_adm_hh_ane $4$ (not in LMFDB) 5.3.ae_j_as_bt_adm $4$ (not in LMFDB) 5.3.ac_d_a_j_as $4$ (not in LMFDB) 5.3.ac_j_am_bt_acc $4$ (not in LMFDB) 5.3.c_d_a_j_s $4$ (not in LMFDB) 5.3.c_j_m_bt_cc $4$ (not in LMFDB) 5.3.e_j_s_bt_dm $4$ (not in LMFDB) 5.3.i_bh_dm_hh_ne $4$ (not in LMFDB) 5.3.f_p_be_bt_cu $5$ (not in LMFDB) 5.3.al_ci_aif_vd_abpo $6$ (not in LMFDB) 5.3.ai_bk_aek_kt_auu $6$ (not in LMFDB) 5.3.af_m_ap_j_a $6$ (not in LMFDB) 5.3.af_v_aci_fo_akk $6$ (not in LMFDB) 5.3.ac_g_ag_j_a $6$ (not in LMFDB) 5.3.ac_p_ay_dm_aee $6$ (not in LMFDB) 5.3.b_a_d_j_a $6$ (not in LMFDB) 5.3.b_j_m_bk_cc $6$ (not in LMFDB) 5.3.e_m_be_cl_ee $6$ (not in LMFDB) 5.3.h_y_cf_en_ii $6$ (not in LMFDB) 5.3.ac_d_a_aj_s $8$ (not in LMFDB) 5.3.c_d_a_aj_as $8$ (not in LMFDB) 5.3.ab_a_am_s_a $9$ (not in LMFDB) 5.3.ab_a_g_a_a $9$ (not in LMFDB) 5.3.c_g_ad_aj_acc $9$ (not in LMFDB) 5.3.c_g_p_bb_cc $9$ (not in LMFDB) 5.3.f_m_g_abk_aee $9$ (not in LMFDB) 5.3.f_m_y_cc_ee $9$ (not in LMFDB) 5.3.af_p_abe_bt_acu $10$ (not in LMFDB) 5.3.b_d_g_j_bk $10$ (not in LMFDB) 5.3.ai_y_as_add_js $12$ (not in LMFDB) 5.3.af_j_a_abk_dm $12$ (not in LMFDB) 5.3.af_s_abt_dv_agy $12$ (not in LMFDB) 5.3.ae_a_s_aj_abk $12$ (not in LMFDB) 5.3.ac_aj_y_s_aee $12$ (not in LMFDB) 5.3.ac_ag_s_j_acu $12$ (not in LMFDB) 5.3.ac_a_g_aj_a $12$ (not in LMFDB) 5.3.ac_d_a_as_bk $12$ (not in LMFDB) 5.3.ac_m_as_cl_acu $12$ (not in LMFDB) 5.3.ab_ad_a_a_s $12$ (not in LMFDB) 5.3.ab_g_aj_bb_abk $12$ (not in LMFDB) 5.3.b_ad_a_a_as $12$ (not in LMFDB) 5.3.b_g_j_bb_bk $12$ (not in LMFDB) 5.3.c_aj_ay_s_ee $12$ (not in LMFDB) 5.3.c_ag_as_j_cu $12$ (not in LMFDB) 5.3.c_a_ag_aj_a $12$ (not in LMFDB) 5.3.c_d_a_as_abk $12$ (not in LMFDB) 5.3.c_m_s_cl_cu $12$ (not in LMFDB) 5.3.e_a_as_aj_bk $12$ (not in LMFDB) 5.3.f_j_a_abk_adm $12$ (not in LMFDB) 5.3.f_s_bt_dv_gy $12$ (not in LMFDB) 5.3.i_y_s_add_ajs $12$ (not in LMFDB) 5.3.ab_d_ag_j_abk $15$ (not in LMFDB) 5.3.c_a_ag_a_s $15$ (not in LMFDB) 5.3.af_m_ay_cc_aee $18$ (not in LMFDB) 5.3.af_m_ag_abk_ee $18$ (not in LMFDB) 5.3.ac_g_ap_bb_acc $18$ (not in LMFDB) 5.3.ac_g_d_aj_cc $18$ (not in LMFDB) 5.3.ab_a_am_s_a $18$ (not in LMFDB) 5.3.b_a_ag_a_a $18$ (not in LMFDB) 5.3.b_a_m_s_a $18$ (not in LMFDB) 5.3.ai_be_aco_dv_afo $24$ (not in LMFDB) 5.3.af_p_abe_cc_adm $24$ (not in LMFDB) 5.3.ae_g_ag_bb_acu $24$ (not in LMFDB) 5.3.ac_ad_m_a_abk $24$ (not in LMFDB) 5.3.ac_a_g_j_abk $24$ (not in LMFDB) 5.3.ac_d_a_s_abk $24$ (not in LMFDB) 5.3.ac_g_ag_bb_abk $24$ (not in LMFDB) 5.3.ac_j_am_bk_abk $24$ (not in LMFDB) 5.3.ab_d_ag_s_as $24$ (not in LMFDB) 5.3.b_d_g_s_s $24$ (not in LMFDB) 5.3.c_ad_am_a_bk $24$ (not in LMFDB) 5.3.c_a_ag_j_bk $24$ (not in LMFDB) 5.3.c_d_a_s_bk $24$ (not in LMFDB) 5.3.c_g_g_bb_bk $24$ (not in LMFDB) 5.3.c_j_m_bk_bk $24$ (not in LMFDB) 5.3.e_g_g_bb_cu $24$ (not in LMFDB) 5.3.f_p_be_cc_dm $24$ (not in LMFDB) 5.3.i_be_co_dv_fo $24$ (not in LMFDB) 5.3.ac_a_g_a_as $30$ (not in LMFDB) 5.3.ac_d_a_a_a $48$ (not in LMFDB) 5.3.c_d_a_a_a $48$ (not in LMFDB) 5.3.ac_g_ag_s_as $60$ (not in LMFDB) 5.3.c_g_g_s_s $60$ (not in LMFDB)