# Properties

 Label 5.3.al_ch_ahx_tz_abna 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: $( 1 - 2 x + 3 x^{2} )( 1 - 3 x + 3 x^{2} )^{2}( 1 - 3 x + 5 x^{2} - 9 x^{3} + 9 x^{4} )$ Frobenius angles: $\pm0.0975263560046$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.304086723985$, $\pm0.527857038681$ 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 $A(\F_{q^r})$ 6 47628 16355808 3927976416 1109870943456 243046260108288 51707352610402278 12401985480130786176 3053557949789098356192 727582255708463853401088

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -7 7 29 91 308 853 2261 6691 20333 59842

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ad 2 $\times$ 1.3.ac $\times$ 2.3.ad_f 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}($$$\Q(\sqrt{-3})$$$)$ 1.3.ac : $$\Q(\sqrt{-2})$$. 2.3.ad_f : 4.0.2197.1.
Endomorphism algebra over $\overline{\F}_{3}$
 The base change of $A$ to $\F_{3^{6}}$ is 1.729.abu $\times$ 1.729.cc 2 $\times$ 2.729.cj_ddt. The endomorphism algebra for each factor is: 1.729.abu : $$\Q(\sqrt{-2})$$. 1.729.cc 2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over $$\Q$$ ramified at $3$ and $\infty$. 2.729.cj_ddt : 4.0.2197.1.
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 $\times$ 1.9.c $\times$ 2.9.b_al. The endomorphism algebra for each factor is: 1.9.ad 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$ 1.9.c : $$\Q(\sqrt{-2})$$. 2.9.b_al : 4.0.2197.1.
• Endomorphism algebra over $\F_{3^{3}}$  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a 2 $\times$ 1.27.k $\times$ 2.27.aj_ct. The endomorphism algebra for each factor is: 1.27.a 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$ 1.27.k : $$\Q(\sqrt{-2})$$. 2.27.aj_ct : 4.0.2197.1.

## 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.ah_x_acb_eh_aic $2$ (not in LMFDB) 5.3.af_l_an_j_ag $2$ (not in LMFDB) 5.3.af_l_ah_av_co $2$ (not in LMFDB) 5.3.ab_ab_af_j_g $2$ (not in LMFDB) 5.3.ab_ab_b_d_g $2$ (not in LMFDB) 5.3.b_ab_ab_d_ag $2$ (not in LMFDB) 5.3.b_ab_f_j_ag $2$ (not in LMFDB) 5.3.f_l_h_av_aco $2$ (not in LMFDB) 5.3.f_l_n_j_g $2$ (not in LMFDB) 5.3.h_x_cb_eh_ic $2$ (not in LMFDB) 5.3.l_ch_hx_tz_bna $2$ (not in LMFDB) 5.3.ai_bj_aef_ke_atq $3$ (not in LMFDB) 5.3.af_l_an_j_ag $3$ (not in LMFDB) 5.3.af_u_acg_ff_ajy $3$ (not in LMFDB) 5.3.ac_f_ah_g_ag $3$ (not in LMFDB) 5.3.b_ab_ab_d_ag $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 5.3.ah_x_acb_eh_aic $2$ (not in LMFDB) 5.3.af_l_an_j_ag $2$ (not in LMFDB) 5.3.af_l_ah_av_co $2$ (not in LMFDB) 5.3.ab_ab_af_j_g $2$ (not in LMFDB) 5.3.ab_ab_b_d_g $2$ (not in LMFDB) 5.3.b_ab_ab_d_ag $2$ (not in LMFDB) 5.3.b_ab_f_j_ag $2$ (not in LMFDB) 5.3.f_l_h_av_aco $2$ (not in LMFDB) 5.3.f_l_n_j_g $2$ (not in LMFDB) 5.3.h_x_cb_eh_ic $2$ (not in LMFDB) 5.3.l_ch_hx_tz_bna $2$ (not in LMFDB) 5.3.ai_bj_aef_ke_atq $3$ (not in LMFDB) 5.3.af_l_an_j_ag $3$ (not in LMFDB) 5.3.af_u_acg_ff_ajy $3$ (not in LMFDB) 5.3.ac_f_ah_g_ag $3$ (not in LMFDB) 5.3.b_ab_ab_d_ag $3$ (not in LMFDB) 5.3.af_r_abr_dp_ags $4$ (not in LMFDB) 5.3.ab_f_al_v_abq $4$ (not in LMFDB) 5.3.b_f_l_v_bq $4$ (not in LMFDB) 5.3.f_r_br_dp_gs $4$ (not in LMFDB) 5.3.ae_l_abd_ci_ady $6$ (not in LMFDB) 5.3.ac_f_ab_ag_be $6$ (not in LMFDB) 5.3.ab_i_ao_bb_aco $6$ (not in LMFDB) 5.3.b_i_o_bb_co $6$ (not in LMFDB) 5.3.c_f_b_ag_abe $6$ (not in LMFDB) 5.3.c_f_h_g_g $6$ (not in LMFDB) 5.3.e_l_bd_ci_dy $6$ (not in LMFDB) 5.3.f_u_cg_ff_jy $6$ (not in LMFDB) 5.3.i_bj_ef_ke_tq $6$ (not in LMFDB) 5.3.af_i_c_abh_da $12$ (not in LMFDB) 5.3.ab_ae_ac_d_be $12$ (not in LMFDB) 5.3.b_ae_c_d_abe $12$ (not in LMFDB) 5.3.f_i_ac_abh_ada $12$ (not in LMFDB) 5.3.af_o_abc_bz_adm $24$ (not in LMFDB) 5.3.ab_c_ai_p_as $24$ (not in LMFDB) 5.3.b_c_i_p_s $24$ (not in LMFDB) 5.3.f_o_bc_bz_dm $24$ (not in LMFDB)