# Properties

 Label 5.3.al_cg_aho_sp_abjx 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: $( 1 - 3 x + 3 x^{2} )^{2}( 1 - 5 x + 13 x^{2} - 25 x^{3} + 39 x^{4} - 45 x^{5} + 27 x^{6} )$ Frobenius angles: $\pm0.0714477711956$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.272071776080$, $\pm0.560185743604$ 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 $A(\F_{q^r})$ 5 37975 12822320 4011109375 1172505761275 235283417286400 49582326048864320 12257075126212109375 3008478155738955147920 717620959298433295774375

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -7 5 23 93 323 827 2170 6613 20039 59025

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ad 2 $\times$ 3.3.af_n_az 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.af_n_az : 6.0.1342367.1.
Endomorphism algebra over $\overline{\F}_{3}$
 The base change of $A$ to $\F_{3^{6}}$ is 1.729.cc 2 $\times$ 3.729.al_cmn_aoxb. 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.al_cmn_aoxb : 6.0.1342367.1.
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.b_ad_ah. The endomorphism algebra for each factor is: 1.9.ad 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$ 3.9.b_ad_ah : 6.0.1342367.1.
• Endomorphism algebra over $\F_{3^{3}}$  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a 2 $\times$ 3.27.af_h_ev. The endomorphism algebra for each factor is: 1.27.a 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$ 3.27.af_h_ev : 6.0.1342367.1.

## 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.ab_ac_e_d_ad $2$ (not in LMFDB) 5.3.f_k_k_j_p $2$ (not in LMFDB) 5.3.l_cg_ho_sp_bjx $2$ (not in LMFDB) 5.3.ai_bi_adz_jm_asg $3$ (not in LMFDB) 5.3.af_k_ak_j_ap $3$ (not in LMFDB) 5.3.af_t_acd_ew_ajg $3$ (not in LMFDB) 5.3.ac_e_ah_g_ag $3$ (not in LMFDB) 5.3.b_ac_ae_d_d $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 5.3.ab_ac_e_d_ad $2$ (not in LMFDB) 5.3.f_k_k_j_p $2$ (not in LMFDB) 5.3.l_cg_ho_sp_bjx $2$ (not in LMFDB) 5.3.ai_bi_adz_jm_asg $3$ (not in LMFDB) 5.3.af_k_ak_j_ap $3$ (not in LMFDB) 5.3.af_t_acd_ew_ajg $3$ (not in LMFDB) 5.3.ac_e_ah_g_ag $3$ (not in LMFDB) 5.3.b_ac_ae_d_d $3$ (not in LMFDB) 5.3.af_q_abo_dj_agj $4$ (not in LMFDB) 5.3.f_q_bo_dj_gj $4$ (not in LMFDB) 5.3.ac_e_ah_g_ag $6$ (not in LMFDB) 5.3.c_e_h_g_g $6$ (not in LMFDB) 5.3.f_t_cd_ew_jg $6$ (not in LMFDB) 5.3.i_bi_dz_jm_sg $6$ (not in LMFDB) 5.3.af_h_f_abe_ci $12$ (not in LMFDB) 5.3.f_h_af_abe_aci $12$ (not in LMFDB) 5.3.af_n_az_bw_adm $24$ (not in LMFDB) 5.3.f_n_z_bw_dm $24$ (not in LMFDB)