# Properties

 Label 5.3.ak_bu_aez_kb_arr 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: $( 1 - 3 x + 3 x^{2} )^{3}( 1 - x + x^{2} - 3 x^{3} + 9 x^{4} )$ Frobenius angles: $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.201748855633$, $\pm0.672988571819$ Angle rank: $2$ (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 $A(\F_{q^r})$ 7 36015 12139456 6725621175 1406550257392 251828644515840 58062531125378017 12624038192963754375 2878843942948456493248 707130482388891369043200

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -6 2 21 134 369 881 2514 6806 19173 58157

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ad 3 $\times$ 2.3.ab_b 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.ab_b : 4.0.16317.1.
Endomorphism algebra over $\overline{\F}_{3}$
 The base change of $A$ to $\F_{3^{6}}$ is 1.729.cc 3 $\times$ 2.729.al_abfv. 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.al_abfv : 4.0.16317.1.
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 3 $\times$ 2.9.b_n. The endomorphism algebra for each factor is: 1.9.ad 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-3})$$$)$ 2.9.b_n : 4.0.16317.1.
• Endomorphism algebra over $\F_{3^{3}}$  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a 3 $\times$ 2.27.ah_t. The endomorphism algebra for each factor is: 1.27.a 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-3})$$$)$ 2.27.ah_t : 4.0.16317.1.

## 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.ai_bc_abz_bt_abb $2$ (not in LMFDB) 5.3.ae_e_d_j_abt $2$ (not in LMFDB) 5.3.ac_ac_j_j_abt $2$ (not in LMFDB) 5.3.c_ac_aj_j_bt $2$ (not in LMFDB) 5.3.e_e_ad_j_bt $2$ (not in LMFDB) 5.3.i_bc_bz_bt_bb $2$ (not in LMFDB) 5.3.k_bu_ez_kb_rr $2$ (not in LMFDB) 5.3.ah_z_acl_ff_ajs $3$ (not in LMFDB) 5.3.ae_e_d_j_abt $3$ (not in LMFDB) 5.3.ae_n_abh_cu_aew $3$ (not in LMFDB) 5.3.ab_b_ad_j_a $3$ (not in LMFDB) 5.3.ab_k_am_bt_acc $3$ (not in LMFDB) 5.3.c_ac_aj_j_bt $3$ (not in LMFDB) 5.3.c_h_j_s_s $3$ (not in LMFDB) 5.3.f_n_v_bb_bk $3$ (not in LMFDB) 5.3.i_bc_bz_bt_bb $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 5.3.ai_bc_abz_bt_abb $2$ (not in LMFDB) 5.3.ae_e_d_j_abt $2$ (not in LMFDB) 5.3.ac_ac_j_j_abt $2$ (not in LMFDB) 5.3.c_ac_aj_j_bt $2$ (not in LMFDB) 5.3.e_e_ad_j_bt $2$ (not in LMFDB) 5.3.i_bc_bz_bt_bb $2$ (not in LMFDB) 5.3.k_bu_ez_kb_rr $2$ (not in LMFDB) 5.3.ah_z_acl_ff_ajs $3$ (not in LMFDB) 5.3.ae_e_d_j_abt $3$ (not in LMFDB) 5.3.ae_n_abh_cu_aew $3$ (not in LMFDB) 5.3.ab_b_ad_j_a $3$ (not in LMFDB) 5.3.ab_k_am_bt_acc $3$ (not in LMFDB) 5.3.c_ac_aj_j_bt $3$ (not in LMFDB) 5.3.c_h_j_s_s $3$ (not in LMFDB) 5.3.f_n_v_bb_bk $3$ (not in LMFDB) 5.3.i_bc_bz_bt_bb $3$ (not in LMFDB) 5.3.ae_k_av_bz_adv $4$ (not in LMFDB) 5.3.ac_e_ad_p_abb $4$ (not in LMFDB) 5.3.c_e_d_p_bb $4$ (not in LMFDB) 5.3.e_k_v_bz_dv $4$ (not in LMFDB) 5.3.af_n_av_bb_abk $6$ (not in LMFDB) 5.3.ac_h_aj_s_as $6$ (not in LMFDB) 5.3.b_b_d_j_a $6$ (not in LMFDB) 5.3.b_k_m_bt_cc $6$ (not in LMFDB) 5.3.e_n_bh_cu_ew $6$ (not in LMFDB) 5.3.h_z_cl_ff_js $6$ (not in LMFDB) 5.3.ab_b_am_s_aj $9$ (not in LMFDB) 5.3.ab_b_g_a_j $9$ (not in LMFDB) 5.3.ae_b_p_am_as $12$ (not in LMFDB) 5.3.ac_af_p_g_acc $12$ (not in LMFDB) 5.3.ab_ac_a_ad_s $12$ (not in LMFDB) 5.3.ab_h_aj_bh_abk $12$ (not in LMFDB) 5.3.b_ac_a_ad_as $12$ (not in LMFDB) 5.3.b_h_j_bh_bk $12$ (not in LMFDB) 5.3.c_af_ap_g_cc $12$ (not in LMFDB) 5.3.e_b_ap_am_s $12$ (not in LMFDB) 5.3.b_b_ag_a_aj $18$ (not in LMFDB) 5.3.b_b_m_s_j $18$ (not in LMFDB) 5.3.ae_h_aj_be_acu $24$ (not in LMFDB) 5.3.ac_b_d_m_abk $24$ (not in LMFDB) 5.3.ab_e_ag_v_as $24$ (not in LMFDB) 5.3.b_e_g_v_s $24$ (not in LMFDB) 5.3.c_b_ad_m_bk $24$ (not in LMFDB) 5.3.e_h_j_be_cu $24$ (not in LMFDB)