# Properties

 Label 5.3.ak_bx_agc_or_abce 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: $( 1 - 3 x + 3 x^{2} )^{2}( 1 - 4 x + 10 x^{2} - 20 x^{3} + 30 x^{4} - 36 x^{5} + 27 x^{6} )$ Frobenius angles: $\pm0.0844416807585$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.360432408976$, $\pm0.575465777728$ 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 $A(\F_{q^r})$ 8 50176 13121024 3697168384 1036176718888 226476012077056 52415886206779096 13023712682309009408 3052799073160079161856 711112415711367293510656

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -6 8 24 84 294 800 2290 7012 20328 58488

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ad 2 $\times$ 3.3.ae_k_au 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_k_au : 6.0.2296688.1.
Endomorphism algebra over $\overline{\F}_{3}$
 The base change of $A$ to $\F_{3^{6}}$ is 1.729.cc 2 $\times$ 3.729.abm_cpz_adepg. 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.abm_cpz_adepg : 6.0.2296688.1.
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.e_a_abi. The endomorphism algebra for each factor is: 1.9.ad 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$ 3.9.e_a_abi : 6.0.2296688.1.
• Endomorphism algebra over $\F_{3^{3}}$  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a 2 $\times$ 3.27.ae_al_ku. The endomorphism algebra for each factor is: 1.27.a 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$ 3.27.ae_al_ku : 6.0.2296688.1.

## 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.ac_b_c_ad_m $2$ (not in LMFDB) 5.3.e_h_i_j_m $2$ (not in LMFDB) 5.3.k_bx_gc_or_bce $2$ (not in LMFDB) 5.3.ah_bc_adf_hn_aoi $3$ (not in LMFDB) 5.3.ae_h_ai_j_am $3$ (not in LMFDB) 5.3.ae_q_abs_dv_ahk $3$ (not in LMFDB) 5.3.ab_e_af_d_am $3$ (not in LMFDB) 5.3.c_b_ac_ad_am $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 5.3.ac_b_c_ad_m $2$ (not in LMFDB) 5.3.e_h_i_j_m $2$ (not in LMFDB) 5.3.k_bx_gc_or_bce $2$ (not in LMFDB) 5.3.ah_bc_adf_hn_aoi $3$ (not in LMFDB) 5.3.ae_h_ai_j_am $3$ (not in LMFDB) 5.3.ae_q_abs_dv_ahk $3$ (not in LMFDB) 5.3.ab_e_af_d_am $3$ (not in LMFDB) 5.3.c_b_ac_ad_am $3$ (not in LMFDB) 5.3.ae_n_abg_cr_afc $4$ (not in LMFDB) 5.3.e_n_bg_cr_fc $4$ (not in LMFDB) 5.3.ah_bc_adf_hn_aoi $6$ (not in LMFDB) 5.3.b_e_f_d_m $6$ (not in LMFDB) 5.3.e_q_bs_dv_hk $6$ (not in LMFDB) 5.3.h_bc_df_hn_oi $6$ (not in LMFDB) 5.3.ae_e_e_av_bw $12$ (not in LMFDB) 5.3.e_e_ae_av_abw $12$ (not in LMFDB) 5.3.ae_k_au_bn_acu $24$ (not in LMFDB) 5.3.e_k_u_bn_cu $24$ (not in LMFDB)