# Properties

 Label 5.3.al_cj_ain_wh_absc 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: $( 1 - 2 x + 3 x^{2} )( 1 - 3 x + 3 x^{2} )^{2}( 1 - 3 x + 7 x^{2} - 9 x^{3} + 9 x^{4} )$ Frobenius angles: $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.227267020856$, $\pm0.304086723985$, $\pm0.464830336654$ Angle rank: $3$ (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 $A(\F_{q^r})$ 10 85260 31430560 5187218400 1101908764000 244394982558720 52712767854495010 11956788913150972800 2906332605830394821920 716644154693490866304000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -7 11 47 115 308 857 2303 6451 19361 58946

## 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.ad_h 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.ad_h : 4.0.1525.1.
Endomorphism algebra over $\overline{\F}_{3}$
 The base change of $A$ to $\F_{3^{6}}$ is 1.729.abu $\times$ 1.729.cc 2 $\times$ 2.729.cn_dov. 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.cn_dov : 4.0.1525.1.
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 2 $\times$ 1.9.c $\times$ 2.9.f_n. 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.f_n : 4.0.1525.1.
• 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.j_cv. 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.j_cv : 4.0.1525.1.

## 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_z_acj_et_aio $2$ (not in LMFDB) 5.3.af_n_ax_bn_aco $2$ (not in LMFDB) 5.3.af_n_ar_j_g $2$ (not in LMFDB) 5.3.ab_b_ah_p_ag $2$ (not in LMFDB) 5.3.ab_b_ab_j_ag $2$ (not in LMFDB) 5.3.b_b_b_j_g $2$ (not in LMFDB) 5.3.b_b_h_p_g $2$ (not in LMFDB) 5.3.f_n_r_j_ag $2$ (not in LMFDB) 5.3.f_n_x_bn_co $2$ (not in LMFDB) 5.3.h_z_cj_et_io $2$ (not in LMFDB) 5.3.l_cj_in_wh_bsc $2$ (not in LMFDB) 5.3.ai_bl_aep_li_avy $3$ (not in LMFDB) 5.3.af_n_ar_j_g $3$ (not in LMFDB) 5.3.af_w_ack_fx_akw $3$ (not in LMFDB) 5.3.ac_h_af_m_g $3$ (not in LMFDB) 5.3.b_b_h_p_g $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 5.3.ah_z_acj_et_aio $2$ (not in LMFDB) 5.3.af_n_ax_bn_aco $2$ (not in LMFDB) 5.3.af_n_ar_j_g $2$ (not in LMFDB) 5.3.ab_b_ah_p_ag $2$ (not in LMFDB) 5.3.ab_b_ab_j_ag $2$ (not in LMFDB) 5.3.b_b_b_j_g $2$ (not in LMFDB) 5.3.b_b_h_p_g $2$ (not in LMFDB) 5.3.f_n_r_j_ag $2$ (not in LMFDB) 5.3.f_n_x_bn_co $2$ (not in LMFDB) 5.3.h_z_cj_et_io $2$ (not in LMFDB) 5.3.l_cj_in_wh_bsc $2$ (not in LMFDB) 5.3.ai_bl_aep_li_avy $3$ (not in LMFDB) 5.3.af_n_ar_j_g $3$ (not in LMFDB) 5.3.af_w_ack_fx_akw $3$ (not in LMFDB) 5.3.ac_h_af_m_g $3$ (not in LMFDB) 5.3.b_b_h_p_g $3$ (not in LMFDB) 5.3.af_t_abv_eb_ahe $4$ (not in LMFDB) 5.3.ab_h_ah_bh_abe $4$ (not in LMFDB) 5.3.b_h_h_bh_be $4$ (not in LMFDB) 5.3.f_t_bv_eb_he $4$ (not in LMFDB) 5.3.ae_n_abf_co_aek $6$ (not in LMFDB) 5.3.ac_h_al_y_abe $6$ (not in LMFDB) 5.3.ab_k_ak_bt_abq $6$ (not in LMFDB) 5.3.b_k_k_bt_bq $6$ (not in LMFDB) 5.3.c_h_f_m_ag $6$ (not in LMFDB) 5.3.c_h_l_y_be $6$ (not in LMFDB) 5.3.e_n_bf_co_ek $6$ (not in LMFDB) 5.3.f_w_ck_fx_kw $6$ (not in LMFDB) 5.3.i_bl_ep_li_vy $6$ (not in LMFDB) 5.3.af_k_ac_abn_dy $12$ (not in LMFDB) 5.3.ab_ac_c_ad_g $12$ (not in LMFDB) 5.3.b_ac_ac_ad_ag $12$ (not in LMFDB) 5.3.f_k_c_abn_ady $12$ (not in LMFDB) 5.3.af_q_abg_cf_adm $24$ (not in LMFDB) 5.3.ab_e_ae_v_as $24$ (not in LMFDB) 5.3.b_e_e_v_s $24$ (not in LMFDB) 5.3.f_q_bg_cf_dm $24$ (not in LMFDB)