# Properties

 Label 5.3.ak_bw_aft_nh_azb 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: $( 1 - 3 x + 3 x^{2} )^{2}( 1 - 4 x + 9 x^{2} - 17 x^{3} + 27 x^{4} - 36 x^{5} + 27 x^{6} )$ Frobenius angles: $\pm0.0653366913680$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.328985474983$, $\pm0.609104440316$ 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 $A(\F_{q^r})$ 7 41503 11217472 4159306151 1059191953232 208570315706368 50778450716647997 12902146522011609863 3010965622576573221952 712713029971932025127168

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -6 6 21 94 299 741 2220 6950 20055 58621

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ad 2 $\times$ 3.3.ae_j_ar 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}($$$\Q(\sqrt{-3})$$$)$ 3.3.ae_j_ar : 6.0.10338167.1.
Endomorphism algebra over $\overline{\F}_{3}$
 The base change of $A$ to $\F_{3^{6}}$ is 1.729.cc 2 $\times$ 3.729.adt_hiy_ajlhg. The endomorphism algebra for each factor is: 1.729.cc 2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over $$\Q$$ ramified at $3$ and $\infty$. 3.729.adt_hiy_ajlhg : 6.0.10338167.1.
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 $\times$ 3.9.c_ab_abl. The endomorphism algebra for each factor is: 1.9.ad 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$ 3.9.c_ab_abl : 6.0.10338167.1.
• Endomorphism algebra over $\F_{3^{3}}$  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a 2 $\times$ 3.27.ah_ay_pq. The endomorphism algebra for each factor is: 1.27.a 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$ 3.27.ah_ay_pq : 6.0.10338167.1.

## 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.ae_g_af_j_av $2$ (not in LMFDB) 5.3.ac_a_f_ad_d $2$ (not in LMFDB) 5.3.c_a_af_ad_ad $2$ (not in LMFDB) 5.3.e_g_f_j_v $2$ (not in LMFDB) 5.3.k_bw_ft_nh_zb $2$ (not in LMFDB) 5.3.ah_bb_acz_gv_amy $3$ (not in LMFDB) 5.3.ae_g_af_j_av $3$ (not in LMFDB) 5.3.ae_p_abp_dm_ags $3$ (not in LMFDB) 5.3.ab_d_af_d_am $3$ (not in LMFDB) 5.3.c_a_af_ad_ad $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 5.3.ae_g_af_j_av $2$ (not in LMFDB) 5.3.ac_a_f_ad_d $2$ (not in LMFDB) 5.3.c_a_af_ad_ad $2$ (not in LMFDB) 5.3.e_g_f_j_v $2$ (not in LMFDB) 5.3.k_bw_ft_nh_zb $2$ (not in LMFDB) 5.3.ah_bb_acz_gv_amy $3$ (not in LMFDB) 5.3.ae_g_af_j_av $3$ (not in LMFDB) 5.3.ae_p_abp_dm_ags $3$ (not in LMFDB) 5.3.ab_d_af_d_am $3$ (not in LMFDB) 5.3.c_a_af_ad_ad $3$ (not in LMFDB) 5.3.ae_m_abd_cl_aet $4$ (not in LMFDB) 5.3.e_m_bd_cl_et $4$ (not in LMFDB) 5.3.b_d_f_d_m $6$ (not in LMFDB) 5.3.e_p_bp_dm_gs $6$ (not in LMFDB) 5.3.h_bb_cz_gv_my $6$ (not in LMFDB) 5.3.ae_d_h_as_be $12$ (not in LMFDB) 5.3.e_d_ah_as_abe $12$ (not in LMFDB) 5.3.ae_j_ar_bk_acu $24$ (not in LMFDB) 5.3.e_j_r_bk_cu $24$ (not in LMFDB)