# Properties

 Label 5.3.ak_bv_afj_lr_avj 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: $( 1 - 3 x + 3 x^{2} )^{2}( 1 - 4 x + 8 x^{2} - 13 x^{3} + 24 x^{4} - 36 x^{5} + 27 x^{6} )$ Frobenius angles: $\pm0.102762435325$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.278353759721$, $\pm0.643265352440$ 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 38759 11464432 5430174659 1229650350257 220397560264448 53076707805199099 12768830646189408427 2963011331031393467392 721245798565095685030919

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -6 4 21 116 334 781 2318 6884 19740 59324

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ad 2 $\times$ 3.3.ae_i_an 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_i_an : 6.0.10816643.1.
Endomorphism algebra over $\overline{\F}_{3}$
 The base change of $A$ to $\F_{3^{6}}$ is 1.729.cc 2 $\times$ 3.729.acf_dba_adgnl. 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.acf_dba_adgnl : 6.0.10816643.1.
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.a_i_at. The endomorphism algebra for each factor is: 1.9.ad 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$ 3.9.a_i_at : 6.0.10816643.1.
• Endomorphism algebra over $\F_{3^{3}}$  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a 2 $\times$ 3.27.ah_ae_gf. The endomorphism algebra for each factor is: 1.27.a 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$ 3.27.ah_ae_gf : 6.0.10816643.1.

## 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_f_ab_j_abh $2$ (not in LMFDB) 5.3.ac_ab_h_d_av $2$ (not in LMFDB) 5.3.c_ab_ah_d_v $2$ (not in LMFDB) 5.3.e_f_b_j_bh $2$ (not in LMFDB) 5.3.k_bv_fj_lr_vj $2$ (not in LMFDB) 5.3.ah_ba_acs_ga_ali $3$ (not in LMFDB) 5.3.ae_o_abl_dd_afu $3$ (not in LMFDB) 5.3.ab_c_ae_g_ag $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 5.3.ae_f_ab_j_abh $2$ (not in LMFDB) 5.3.ac_ab_h_d_av $2$ (not in LMFDB) 5.3.c_ab_ah_d_v $2$ (not in LMFDB) 5.3.e_f_b_j_bh $2$ (not in LMFDB) 5.3.k_bv_fj_lr_vj $2$ (not in LMFDB) 5.3.ah_ba_acs_ga_ali $3$ (not in LMFDB) 5.3.ae_o_abl_dd_afu $3$ (not in LMFDB) 5.3.ab_c_ae_g_ag $3$ (not in LMFDB) 5.3.ae_l_az_cf_aeh $4$ (not in LMFDB) 5.3.e_l_z_cf_eh $4$ (not in LMFDB) 5.3.ac_ab_h_d_av $6$ (not in LMFDB) 5.3.b_c_e_g_g $6$ (not in LMFDB) 5.3.e_o_bl_dd_fu $6$ (not in LMFDB) 5.3.h_ba_cs_ga_li $6$ (not in LMFDB) 5.3.ae_c_l_ap_g $12$ (not in LMFDB) 5.3.e_c_al_ap_ag $12$ (not in LMFDB) 5.3.ae_i_an_bh_acu $24$ (not in LMFDB) 5.3.e_i_n_bh_cu $24$ (not in LMFDB)