# Properties

 Label 5.3.ak_bv_afi_ll_auu 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: $( 1 - 3 x + 3 x^{2} )^{3}( 1 - x + 2 x^{2} - 3 x^{3} + 9 x^{4} )$ Frobenius angles: $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.235082516458$, $\pm0.648854628963$ 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 $A(\F_{q^r})$ 8 43904 13346816 6559081984 1434732212968 243767101620224 55176829474152856 12584463561909098496 2879291214277250275328 706286311054803767437184

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -6 4 24 132 374 856 2402 6788 19176 58084

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ad 3 $\times$ 2.3.ab_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.ab_c : 4.0.3757.1.
Endomorphism algebra over $\overline{\F}_{3}$
 The base change of $A$ to $\F_{3^{6}}$ is 1.729.cc 3 $\times$ 2.729.abk_bas. 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.abk_bas : 4.0.3757.1.
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.d_q. The endomorphism algebra for each factor is: 1.9.ad 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-3})$$$)$ 2.9.d_q : 4.0.3757.1.
• Endomorphism algebra over $\F_{3^{3}}$  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a 3 $\times$ 2.27.ae_ak. The endomorphism algebra for each factor is: 1.27.a 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-3})$$$)$ 2.27.ae_ak : 4.0.3757.1.

## 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_bd_aci_dd_aee $2$ (not in LMFDB) 5.3.ae_f_a_j_abk $2$ (not in LMFDB) 5.3.ac_ab_g_j_abk $2$ (not in LMFDB) 5.3.c_ab_ag_j_bk $2$ (not in LMFDB) 5.3.e_f_a_j_bk $2$ (not in LMFDB) 5.3.i_bd_ci_dd_ee $2$ (not in LMFDB) 5.3.k_bv_fi_ll_uu $2$ (not in LMFDB) 5.3.ah_ba_acr_fx_alc $3$ (not in LMFDB) 5.3.ae_f_a_j_abk $3$ (not in LMFDB) 5.3.ae_o_abk_dd_afo $3$ (not in LMFDB) 5.3.ab_c_ad_j_a $3$ (not in LMFDB) 5.3.ab_l_am_cc_acc $3$ (not in LMFDB) 5.3.c_ab_ag_j_bk $3$ (not in LMFDB) 5.3.c_i_m_bb_bk $3$ (not in LMFDB) 5.3.f_o_bb_bt_cu $3$ (not in LMFDB) 5.3.i_bd_ci_dd_ee $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 5.3.ai_bd_aci_dd_aee $2$ (not in LMFDB) 5.3.ae_f_a_j_abk $2$ (not in LMFDB) 5.3.ac_ab_g_j_abk $2$ (not in LMFDB) 5.3.c_ab_ag_j_bk $2$ (not in LMFDB) 5.3.e_f_a_j_bk $2$ (not in LMFDB) 5.3.i_bd_ci_dd_ee $2$ (not in LMFDB) 5.3.k_bv_fi_ll_uu $2$ (not in LMFDB) 5.3.ah_ba_acr_fx_alc $3$ (not in LMFDB) 5.3.ae_f_a_j_abk $3$ (not in LMFDB) 5.3.ae_o_abk_dd_afo $3$ (not in LMFDB) 5.3.ab_c_ad_j_a $3$ (not in LMFDB) 5.3.ab_l_am_cc_acc $3$ (not in LMFDB) 5.3.c_ab_ag_j_bk $3$ (not in LMFDB) 5.3.c_i_m_bb_bk $3$ (not in LMFDB) 5.3.f_o_bb_bt_cu $3$ (not in LMFDB) 5.3.i_bd_ci_dd_ee $3$ (not in LMFDB) 5.3.ae_l_ay_cf_aee $4$ (not in LMFDB) 5.3.ac_f_ag_v_abk $4$ (not in LMFDB) 5.3.c_f_g_v_bk $4$ (not in LMFDB) 5.3.e_l_y_cf_ee $4$ (not in LMFDB) 5.3.af_o_abb_bt_acu $6$ (not in LMFDB) 5.3.ac_i_am_bb_abk $6$ (not in LMFDB) 5.3.b_c_d_j_a $6$ (not in LMFDB) 5.3.b_l_m_cc_cc $6$ (not in LMFDB) 5.3.e_o_bk_dd_fo $6$ (not in LMFDB) 5.3.h_ba_cr_fx_lc $6$ (not in LMFDB) 5.3.ab_c_am_s_as $9$ (not in LMFDB) 5.3.ab_c_g_a_s $9$ (not in LMFDB) 5.3.ae_c_m_ap_a $12$ (not in LMFDB) 5.3.ac_ae_m_d_abk $12$ (not in LMFDB) 5.3.ab_ab_a_ag_s $12$ (not in LMFDB) 5.3.ab_i_aj_bn_abk $12$ (not in LMFDB) 5.3.b_ab_a_ag_as $12$ (not in LMFDB) 5.3.b_i_j_bn_bk $12$ (not in LMFDB) 5.3.c_ae_am_d_bk $12$ (not in LMFDB) 5.3.e_c_am_ap_a $12$ (not in LMFDB) 5.3.b_c_ag_a_as $18$ (not in LMFDB) 5.3.b_c_m_s_s $18$ (not in LMFDB) 5.3.ae_i_am_bh_acu $24$ (not in LMFDB) 5.3.ac_c_a_p_abk $24$ (not in LMFDB) 5.3.ab_f_ag_y_as $24$ (not in LMFDB) 5.3.b_f_g_y_s $24$ (not in LMFDB) 5.3.c_c_a_p_bk $24$ (not in LMFDB) 5.3.e_i_m_bh_cu $24$ (not in LMFDB)