# Properties

 Label 5.3.al_ce_agv_pp_abdc 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: $( 1 - 3 x + 3 x^{2} )^{3}( 1 - 2 x + 2 x^{2} - 6 x^{3} + 9 x^{4} )$ Frobenius angles: $\pm0.116139763599$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.616139763599$ 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 $A(\F_{q^r})$ 4 27440 9571072 4822854400 1374148969484 256327090503680 56541384167257364 13238272102480588800 2990708809374622556416 709094956080155504846000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -7 1 14 105 363 892 2457 7121 19922 58321

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ad 3 $\times$ 2.3.ac_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 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-3})$$$)$ 2.3.ac_c : $$\Q(i, \sqrt{5})$$.
Endomorphism algebra over $\overline{\F}_{3}$
 The base change of $A$ to $\F_{3^{12}}$ is 1.531441.acec 3 $\times$ 1.531441.sk 2 . The endomorphism algebra for each factor is: 1.531441.acec 3 : $\mathrm{M}_{3}(B)$, where $B$ is the quaternion algebra over $$\Q$$ ramified at $3$ and $\infty$. 1.531441.sk 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-5})$$$)$
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 3 $\times$ 2.9.a_ac. The endomorphism algebra for each factor is: 1.9.ad 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-3})$$$)$ 2.9.a_ac : $$\Q(i, \sqrt{5})$$.
• Endomorphism algebra over $\F_{3^{3}}$  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a 3 $\times$ 2.27.ao_du. The endomorphism algebra for each factor is: 1.27.a 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-3})$$$)$ 2.27.ao_du : $$\Q(i, \sqrt{5})$$.
• Endomorphism algebra over $\F_{3^{4}}$  The base change of $A$ to $\F_{3^{4}}$ is 1.81.ac 2 $\times$ 1.81.j 3 . The endomorphism algebra for each factor is: 1.81.ac 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-5})$$$)$ 1.81.j 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-3})$$$)$
• Endomorphism algebra over $\F_{3^{6}}$  The base change of $A$ to $\F_{3^{6}}$ is 1.729.cc 3 $\times$ 2.729.a_sk. The endomorphism algebra for each factor is: 1.729.cc 3 : $\mathrm{M}_{3}(B)$, where $B$ is the quaternion algebra over $$\Q$$ ramified at $3$ and $\infty$. 2.729.a_sk : $$\Q(i, \sqrt{5})$$.

## 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.ah_u_av_abb_ee $2$ (not in LMFDB) 5.3.af_i_ad_j_abk $2$ (not in LMFDB) 5.3.ab_ae_j_j_abk $2$ (not in LMFDB) 5.3.b_ae_aj_j_bk $2$ (not in LMFDB) 5.3.f_i_d_j_bk $2$ (not in LMFDB) 5.3.h_u_v_abb_aee $2$ (not in LMFDB) 5.3.l_ce_gv_pp_bdc $2$ (not in LMFDB) 5.3.ai_bg_adm_hz_apg $3$ (not in LMFDB) 5.3.af_i_ad_j_abk $3$ (not in LMFDB) 5.3.af_r_abw_ee_ahq $3$ (not in LMFDB) 5.3.ac_c_ag_j_a $3$ (not in LMFDB) 5.3.ac_l_ay_cc_aee $3$ (not in LMFDB) 5.3.b_ae_aj_j_bk $3$ (not in LMFDB) 5.3.b_f_a_a_as $3$ (not in LMFDB) 5.3.e_i_g_aj_abk $3$ (not in LMFDB) 5.3.h_u_v_abb_aee $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 5.3.ah_u_av_abb_ee $2$ (not in LMFDB) 5.3.af_i_ad_j_abk $2$ (not in LMFDB) 5.3.ab_ae_j_j_abk $2$ (not in LMFDB) 5.3.b_ae_aj_j_bk $2$ (not in LMFDB) 5.3.f_i_d_j_bk $2$ (not in LMFDB) 5.3.h_u_v_abb_aee $2$ (not in LMFDB) 5.3.l_ce_gv_pp_bdc $2$ (not in LMFDB) 5.3.ai_bg_adm_hz_apg $3$ (not in LMFDB) 5.3.af_i_ad_j_abk $3$ (not in LMFDB) 5.3.af_r_abw_ee_ahq $3$ (not in LMFDB) 5.3.ac_c_ag_j_a $3$ (not in LMFDB) 5.3.ac_l_ay_cc_aee $3$ (not in LMFDB) 5.3.b_ae_aj_j_bk $3$ (not in LMFDB) 5.3.b_f_a_a_as $3$ (not in LMFDB) 5.3.e_i_g_aj_abk $3$ (not in LMFDB) 5.3.h_u_v_abb_aee $3$ (not in LMFDB) 5.3.af_o_abh_cx_afo $4$ (not in LMFDB) 5.3.ab_c_d_d_a $4$ (not in LMFDB) 5.3.b_c_ad_d_a $4$ (not in LMFDB) 5.3.f_o_bh_cx_fo $4$ (not in LMFDB) 5.3.ae_i_ag_aj_bk $6$ (not in LMFDB) 5.3.ab_f_a_a_s $6$ (not in LMFDB) 5.3.c_c_g_j_a $6$ (not in LMFDB) 5.3.c_l_y_cc_ee $6$ (not in LMFDB) 5.3.f_r_bw_ee_hq $6$ (not in LMFDB) 5.3.i_bg_dm_hz_pg $6$ (not in LMFDB) 5.3.aj_bg_abt_abb_gg $8$ (not in LMFDB) 5.3.aj_bo_aen_kb_ass $8$ (not in LMFDB) 5.3.ad_ae_v_j_adm $8$ (not in LMFDB) 5.3.ad_c_d_d_as $8$ (not in LMFDB) 5.3.ad_e_ad_j_as $8$ (not in LMFDB) 5.3.ad_k_av_bz_adm $8$ (not in LMFDB) 5.3.d_ae_av_j_dm $8$ (not in LMFDB) 5.3.d_c_ad_d_s $8$ (not in LMFDB) 5.3.d_e_d_j_s $8$ (not in LMFDB) 5.3.d_k_v_bz_dm $8$ (not in LMFDB) 5.3.j_bg_bt_abb_agg $8$ (not in LMFDB) 5.3.j_bo_en_kb_ss $8$ (not in LMFDB) 5.3.ac_c_ap_bb_as $9$ (not in LMFDB) 5.3.ac_c_d_aj_s $9$ (not in LMFDB) 5.3.af_f_m_ay_s $12$ (not in LMFDB) 5.3.ac_ab_a_ag_bk $12$ (not in LMFDB) 5.3.ac_i_as_bn_acu $12$ (not in LMFDB) 5.3.ab_ah_m_m_acc $12$ (not in LMFDB) 5.3.b_ah_am_m_cc $12$ (not in LMFDB) 5.3.c_ab_a_ag_abk $12$ (not in LMFDB) 5.3.c_i_s_bn_cu $12$ (not in LMFDB) 5.3.f_f_am_ay_as $12$ (not in LMFDB) 5.3.c_c_ad_aj_as $18$ (not in LMFDB) 5.3.c_c_p_bb_s $18$ (not in LMFDB) 5.3.ag_o_am_aj_bk $24$ (not in LMFDB) 5.3.ag_w_aci_ff_ajs $24$ (not in LMFDB) 5.3.af_l_as_bq_adm $24$ (not in LMFDB) 5.3.ad_ah_be_m_aew $24$ (not in LMFDB) 5.3.ad_ab_m_g_acc $24$ (not in LMFDB) 5.3.ad_b_g_am_s $24$ (not in LMFDB) 5.3.ad_f_ag_a_s $24$ (not in LMFDB) 5.3.ad_h_am_be_acc $24$ (not in LMFDB) 5.3.ad_n_abe_cu_aew $24$ (not in LMFDB) 5.3.ac_f_am_y_abk $24$ (not in LMFDB) 5.3.ab_ab_g_g_as $24$ (not in LMFDB) 5.3.a_ah_a_m_a $24$ (not in LMFDB) 5.3.a_ae_a_j_a $24$ (not in LMFDB) 5.3.a_ab_a_g_a $24$ (not in LMFDB) 5.3.a_b_a_am_a $24$ (not in LMFDB) 5.3.a_c_a_d_a $24$ (not in LMFDB) 5.3.a_e_a_j_a $24$ (not in LMFDB) 5.3.a_f_a_a_a $24$ (not in LMFDB) 5.3.a_h_a_be_a $24$ (not in LMFDB) 5.3.a_k_a_bz_a $24$ (not in LMFDB) 5.3.a_n_a_cu_a $24$ (not in LMFDB) 5.3.b_ab_ag_g_s $24$ (not in LMFDB) 5.3.c_f_m_y_bk $24$ (not in LMFDB) 5.3.d_ah_abe_m_ew $24$ (not in LMFDB) 5.3.d_ab_am_g_cc $24$ (not in LMFDB) 5.3.d_b_ag_am_as $24$ (not in LMFDB) 5.3.d_f_g_a_as $24$ (not in LMFDB) 5.3.d_h_m_be_cc $24$ (not in LMFDB) 5.3.d_n_be_cu_ew $24$ (not in LMFDB) 5.3.f_l_s_bq_dm $24$ (not in LMFDB) 5.3.g_o_m_aj_abk $24$ (not in LMFDB) 5.3.g_w_ci_ff_js $24$ (not in LMFDB) 5.3.a_ae_aj_j_bk $72$ (not in LMFDB) 5.3.a_ae_j_j_abk $72$ (not in LMFDB) 5.3.a_e_aj_j_abk $72$ (not in LMFDB) 5.3.a_e_j_j_bk $72$ (not in LMFDB)