# Properties

 Label 5.3.ak_bw_afs_nb_aym 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: $( 1 - 2 x + 3 x^{2} )( 1 - 3 x + 3 x^{2} )^{2}( 1 - 2 x + 2 x^{2} - 6 x^{3} + 9 x^{4} )$ Frobenius angles: $\pm0.116139763599$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.304086723985$, $\pm0.616139763599$ Angle rank: $2$ (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 $A(\F_{q^r})$ 8 47040 12989312 5087846400 1227099817768 223632308551680 52379898422025112 13009098341862604800 3019576655024956750976 717837007930413343896000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -6 6 24 110 334 792 2290 7006 20112 59046

## 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_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 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$ 1.3.ac : $$\Q(\sqrt{-2})$$. 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 2 $\times$ 1.531441.azi $\times$ 1.531441.sk 2 . The endomorphism algebra for each factor is: 1.531441.acec 2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over $$\Q$$ ramified at $3$ and $\infty$. 1.531441.azi : $$\Q(\sqrt{-2})$$. 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 2 $\times$ 1.9.c $\times$ 2.9.a_ac. 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.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 2 $\times$ 1.27.k $\times$ 2.27.ao_du. The endomorphism algebra for each factor is: 1.27.a 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$ 1.27.k : $$\Q(\sqrt{-2})$$. 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 2 $\times$ 1.81.o. The endomorphism algebra for each factor is: 1.81.ac 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-5})$$$)$ 1.81.j 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$ 1.81.o : $$\Q(\sqrt{-2})$$.
• Endomorphism algebra over $\F_{3^{6}}$  The base change of $A$ to $\F_{3^{6}}$ is 1.729.abu $\times$ 1.729.cc 2 $\times$ 2.729.a_sk. The endomorphism algebra for each factor is: 1.729.abu : $$\Q(\sqrt{-2})$$. 1.729.cc 2 : $\mathrm{M}_{2}(B)$, 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.ag_q_abg_cx_aga $2$ (not in LMFDB) 5.3.ag_q_aq_av_dg $2$ (not in LMFDB) 5.3.ae_g_ae_j_ay $2$ (not in LMFDB) 5.3.ac_a_e_d_am $2$ (not in LMFDB) 5.3.a_ac_ai_j_y $2$ (not in LMFDB) 5.3.a_ac_i_j_ay $2$ (not in LMFDB) 5.3.c_a_ae_d_m $2$ (not in LMFDB) 5.3.e_g_e_j_y $2$ (not in LMFDB) 5.3.g_q_q_av_adg $2$ (not in LMFDB) 5.3.g_q_bg_cx_ga $2$ (not in LMFDB) 5.3.k_bw_fs_nb_ym $2$ (not in LMFDB) 5.3.ah_bb_acy_gs_ams $3$ (not in LMFDB) 5.3.ae_p_abo_dm_agm $3$ (not in LMFDB) 5.3.ab_d_ae_g_ag $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 5.3.ag_q_abg_cx_aga $2$ (not in LMFDB) 5.3.ag_q_aq_av_dg $2$ (not in LMFDB) 5.3.ae_g_ae_j_ay $2$ (not in LMFDB) 5.3.ac_a_e_d_am $2$ (not in LMFDB) 5.3.a_ac_ai_j_y $2$ (not in LMFDB) 5.3.a_ac_i_j_ay $2$ (not in LMFDB) 5.3.c_a_ae_d_m $2$ (not in LMFDB) 5.3.e_g_e_j_y $2$ (not in LMFDB) 5.3.g_q_q_av_adg $2$ (not in LMFDB) 5.3.g_q_bg_cx_ga $2$ (not in LMFDB) 5.3.k_bw_fs_nb_ym $2$ (not in LMFDB) 5.3.ah_bb_acy_gs_ams $3$ (not in LMFDB) 5.3.ae_p_abo_dm_agm $3$ (not in LMFDB) 5.3.ab_d_ae_g_ag $3$ (not in LMFDB) 5.3.ae_m_abc_cl_aeq $4$ (not in LMFDB) 5.3.a_e_ai_p_ay $4$ (not in LMFDB) 5.3.a_e_i_p_y $4$ (not in LMFDB) 5.3.e_m_bc_cl_eq $4$ (not in LMFDB) 5.3.ad_h_au_bq_aco $6$ (not in LMFDB) 5.3.ad_h_ae_ag_be $6$ (not in LMFDB) 5.3.a_h_ai_s_abw $6$ (not in LMFDB) 5.3.a_h_i_s_bw $6$ (not in LMFDB) 5.3.b_d_e_g_g $6$ (not in LMFDB) 5.3.d_h_e_ag_abe $6$ (not in LMFDB) 5.3.d_h_u_bq_co $6$ (not in LMFDB) 5.3.e_p_bo_dm_gm $6$ (not in LMFDB) 5.3.h_bb_cy_gs_ms $6$ (not in LMFDB) 5.3.ai_ba_abi_av_eq $8$ (not in LMFDB) 5.3.ai_bi_adu_il_aps $8$ (not in LMFDB) 5.3.ae_c_k_d_abw $8$ (not in LMFDB) 5.3.ae_k_aw_bz_ads $8$ (not in LMFDB) 5.3.ac_ae_o_j_aci $8$ (not in LMFDB) 5.3.ac_c_c_d_am $8$ (not in LMFDB) 5.3.ac_e_ac_j_am $8$ (not in LMFDB) 5.3.ac_k_ao_bz_aci $8$ (not in LMFDB) 5.3.c_ae_ao_j_ci $8$ (not in LMFDB) 5.3.c_c_ac_d_m $8$ (not in LMFDB) 5.3.c_e_c_j_m $8$ (not in LMFDB) 5.3.c_k_o_bz_ci $8$ (not in LMFDB) 5.3.e_c_ak_d_bw $8$ (not in LMFDB) 5.3.e_k_w_bz_ds $8$ (not in LMFDB) 5.3.i_ba_bi_av_aeq $8$ (not in LMFDB) 5.3.i_bi_du_il_ps $8$ (not in LMFDB) 5.3.ae_d_i_as_y $12$ (not in LMFDB) 5.3.a_af_ai_g_bw $12$ (not in LMFDB) 5.3.a_af_i_g_abw $12$ (not in LMFDB) 5.3.e_d_ai_as_ay $12$ (not in LMFDB) 5.3.af_l_ak_ag_be $24$ (not in LMFDB) 5.3.af_t_aby_ek_aic $24$ (not in LMFDB) 5.3.ae_j_aq_bk_acu $24$ (not in LMFDB) 5.3.ac_ah_u_m_adg $24$ (not in LMFDB) 5.3.ac_ab_i_g_abk $24$ (not in LMFDB) 5.3.ac_b_e_am_m $24$ (not in LMFDB) 5.3.ac_f_ae_a_m $24$ (not in LMFDB) 5.3.ac_h_ai_be_abk $24$ (not in LMFDB) 5.3.ac_n_au_cu_adg $24$ (not in LMFDB) 5.3.ab_ab_ac_g_g $24$ (not in LMFDB) 5.3.ab_h_ak_be_abq $24$ (not in LMFDB) 5.3.a_b_ai_m_a $24$ (not in LMFDB) 5.3.a_b_i_m_a $24$ (not in LMFDB) 5.3.b_ab_c_g_ag $24$ (not in LMFDB) 5.3.b_h_k_be_bq $24$ (not in LMFDB) 5.3.c_ah_au_m_dg $24$ (not in LMFDB) 5.3.c_ab_ai_g_bk $24$ (not in LMFDB) 5.3.c_b_ae_am_am $24$ (not in LMFDB) 5.3.c_f_e_a_am $24$ (not in LMFDB) 5.3.c_h_i_be_bk $24$ (not in LMFDB) 5.3.c_n_u_cu_dg $24$ (not in LMFDB) 5.3.e_j_q_bk_cu $24$ (not in LMFDB) 5.3.f_l_k_ag_abe $24$ (not in LMFDB) 5.3.f_t_by_ek_ic $24$ (not in LMFDB)