# Properties

 Label 5.3.ak_bw_afr_mv_axx Base Field $\F_{3}$ Dimension $5$ Ordinary No $p$-rank $1$ Principally polarizable Yes Contains a Jacobian No

## Invariants

 Base field: $\F_{3}$ Dimension: $5$ L-polynomial: $( 1 - 3 x + 3 x^{2} )^{3}( 1 - x + 3 x^{2} - 3 x^{3} + 9 x^{4} )$ Frobenius angles: $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.268536328535$, $\pm0.622727850897$ Angle rank: $2$ (numerical)

This isogeny class is not simple.

## Newton polygon

 $p$-rank: $1$ Slopes: $[0, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1]$

## 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 $A(\F_{q^r})$ 9 52479 14817600 6110707239 1407187137744 237776723251200 52718628794992659 12648916238326409223 2930642897336370619200 707910798769618802690304

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -6 6 27 126 369 837 2304 6822 19521 58221

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ad 3 $\times$ 2.3.ab_d 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_d : 4.0.11661.1.
Endomorphism algebra over $\overline{\F}_{3}$
 The base change of $A$ to $\F_{3^{6}}$ is 1.729.cc 3 $\times$ 2.729.acd_deb. 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.acd_deb : 4.0.11661.1.
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.f_v. The endomorphism algebra for each factor is: 1.9.ad 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-3})$$$)$ 2.9.f_v : 4.0.11661.1.
• Endomorphism algebra over $\F_{3^{3}}$  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a 3 $\times$ 2.27.ab_abb. The endomorphism algebra for each factor is: 1.27.a 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-3})$$$)$ 2.27.ab_abb : 4.0.11661.1.

## 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_be_acr_en_ahh $2$ (not in LMFDB) 5.3.ac_a_d_j_abb $2$ (not in LMFDB) 5.3.e_g_d_j_bb $2$ (not in LMFDB) 5.3.k_bw_fr_mv_xx $2$ (not in LMFDB) 5.3.ah_bb_acx_gp_amm $3$ (not in LMFDB) 5.3.ae_g_ad_j_abb $3$ (not in LMFDB) 5.3.ae_p_abn_dm_agg $3$ (not in LMFDB) 5.3.ab_d_ad_j_a $3$ (not in LMFDB) 5.3.ab_m_am_cl_acc $3$ (not in LMFDB) 5.3.c_a_ad_j_bb $3$ (not in LMFDB) 5.3.c_j_p_bk_cc $3$ (not in LMFDB) 5.3.f_p_bh_cl_ee $3$ (not in LMFDB) 5.3.i_be_cr_en_hh $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 5.3.ai_be_acr_en_ahh $2$ (not in LMFDB) 5.3.ac_a_d_j_abb $2$ (not in LMFDB) 5.3.e_g_d_j_bb $2$ (not in LMFDB) 5.3.k_bw_fr_mv_xx $2$ (not in LMFDB) 5.3.ah_bb_acx_gp_amm $3$ (not in LMFDB) 5.3.ae_g_ad_j_abb $3$ (not in LMFDB) 5.3.ae_p_abn_dm_agg $3$ (not in LMFDB) 5.3.ab_d_ad_j_a $3$ (not in LMFDB) 5.3.ab_m_am_cl_acc $3$ (not in LMFDB) 5.3.c_a_ad_j_bb $3$ (not in LMFDB) 5.3.c_j_p_bk_cc $3$ (not in LMFDB) 5.3.f_p_bh_cl_ee $3$ (not in LMFDB) 5.3.i_be_cr_en_hh $3$ (not in LMFDB) 5.3.ae_m_abb_cl_aen $4$ (not in LMFDB) 5.3.ac_g_aj_bb_abt $4$ (not in LMFDB) 5.3.c_g_j_bb_bt $4$ (not in LMFDB) 5.3.e_m_bb_cl_en $4$ (not in LMFDB) 5.3.af_p_abh_cl_aee $6$ (not in LMFDB) 5.3.ac_j_ap_bk_acc $6$ (not in LMFDB) 5.3.ab_d_ad_j_a $6$ (not in LMFDB) 5.3.b_d_d_j_a $6$ (not in LMFDB) 5.3.b_m_m_cl_cc $6$ (not in LMFDB) 5.3.e_p_bn_dm_gg $6$ (not in LMFDB) 5.3.h_bb_cx_gp_mm $6$ (not in LMFDB) 5.3.ab_d_am_s_abb $9$ (not in LMFDB) 5.3.ab_d_g_a_bb $9$ (not in LMFDB) 5.3.ae_d_j_as_s $12$ (not in LMFDB) 5.3.ac_ad_j_a_as $12$ (not in LMFDB) 5.3.ab_a_a_aj_s $12$ (not in LMFDB) 5.3.ab_j_aj_bt_abk $12$ (not in LMFDB) 5.3.b_a_a_aj_as $12$ (not in LMFDB) 5.3.b_j_j_bt_bk $12$ (not in LMFDB) 5.3.c_ad_aj_a_s $12$ (not in LMFDB) 5.3.e_d_aj_as_as $12$ (not in LMFDB) 5.3.b_d_ag_a_abb $18$ (not in LMFDB) 5.3.b_d_m_s_bb $18$ (not in LMFDB) 5.3.ae_j_ap_bk_acu $24$ (not in LMFDB) 5.3.ac_d_ad_s_abk $24$ (not in LMFDB) 5.3.ab_g_ag_bb_as $24$ (not in LMFDB) 5.3.b_g_g_bb_s $24$ (not in LMFDB) 5.3.c_d_d_s_bk $24$ (not in LMFDB) 5.3.e_j_p_bk_cu $24$ (not in LMFDB)