# Properties

 Label 5.3.ak_by_agl_qb_abfh Base Field $\F_{3}$ Dimension $5$ Ordinary No $p$-rank $3$ Principally polarizable Yes Contains a Jacobian No

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## Invariants

 Base field: $\F_{3}$ Dimension: $5$ L-polynomial: $( 1 - x + 3 x^{2} )( 1 - 3 x + 3 x^{2} )^{2}( 1 - 3 x + 5 x^{2} - 9 x^{3} + 9 x^{4} )$ Frobenius angles: $\pm0.0975263560046$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.406785250661$, $\pm0.527857038681$ 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 $A(\F_{q^r})$ 9 59535 15494976 3068731575 976869879984 255838168535040 55864604080981719 12681258131107995975 3003469397327024990784 715886088626675908972800

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -6 10 27 70 279 889 2430 6838 20007 58885

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ad 2 $\times$ 1.3.ab $\times$ 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.ab : $$\Q(\sqrt{-11})$$. 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.ak $\times$ 1.729.cc 2 $\times$ 2.729.cj_ddt. The endomorphism algebra for each factor is: 1.729.ak : $$\Q(\sqrt{-11})$$. 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.f $\times$ 2.9.b_al. The endomorphism algebra for each factor is: 1.9.ad 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$ 1.9.f : $$\Q(\sqrt{-11})$$. 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.i $\times$ 2.27.aj_ct. The endomorphism algebra for each factor is: 1.27.a 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$ 1.27.i : $$\Q(\sqrt{-11})$$. 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.ai_bg_adn_if_apv $2$ (not in LMFDB) 5.3.ae_i_al_j_ad $2$ (not in LMFDB) 5.3.ae_i_af_ap_bz $2$ (not in LMFDB) 5.3.ac_c_ah_j_d $2$ (not in LMFDB) 5.3.ac_c_ab_ad_v $2$ (not in LMFDB) 5.3.c_c_b_ad_av $2$ (not in LMFDB) 5.3.c_c_h_j_ad $2$ (not in LMFDB) 5.3.e_i_f_ap_abz $2$ (not in LMFDB) 5.3.e_i_l_j_d $2$ (not in LMFDB) 5.3.i_bg_dn_if_pv $2$ (not in LMFDB) 5.3.k_by_gl_qb_bfh $2$ (not in LMFDB) 5.3.ah_bd_adl_if_aps $3$ (not in LMFDB) 5.3.ae_i_al_j_ad $3$ (not in LMFDB) 5.3.ae_r_abv_ee_aic $3$ (not in LMFDB) 5.3.ab_f_af_d_am $3$ (not in LMFDB) 5.3.c_c_b_ad_av $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 5.3.ai_bg_adn_if_apv $2$ (not in LMFDB) 5.3.ae_i_al_j_ad $2$ (not in LMFDB) 5.3.ae_i_af_ap_bz $2$ (not in LMFDB) 5.3.ac_c_ah_j_d $2$ (not in LMFDB) 5.3.ac_c_ab_ad_v $2$ (not in LMFDB) 5.3.c_c_b_ad_av $2$ (not in LMFDB) 5.3.c_c_h_j_ad $2$ (not in LMFDB) 5.3.e_i_f_ap_abz $2$ (not in LMFDB) 5.3.e_i_l_j_d $2$ (not in LMFDB) 5.3.i_bg_dn_if_pv $2$ (not in LMFDB) 5.3.k_by_gl_qb_bfh $2$ (not in LMFDB) 5.3.ah_bd_adl_if_aps $3$ (not in LMFDB) 5.3.ae_i_al_j_ad $3$ (not in LMFDB) 5.3.ae_r_abv_ee_aic $3$ (not in LMFDB) 5.3.ab_f_af_d_am $3$ (not in LMFDB) 5.3.c_c_b_ad_av $3$ (not in LMFDB) 5.3.ae_o_abj_cx_afl $4$ (not in LMFDB) 5.3.ac_i_at_bn_acx $4$ (not in LMFDB) 5.3.c_i_t_bn_cx $4$ (not in LMFDB) 5.3.e_o_bj_cx_fl $4$ (not in LMFDB) 5.3.af_r_abx_eh_ahw $6$ (not in LMFDB) 5.3.ac_l_az_cc_aek $6$ (not in LMFDB) 5.3.ab_f_b_ad_y $6$ (not in LMFDB) 5.3.b_f_ab_ad_ay $6$ (not in LMFDB) 5.3.b_f_f_d_m $6$ (not in LMFDB) 5.3.c_l_z_cc_ek $6$ (not in LMFDB) 5.3.e_r_bv_ee_ic $6$ (not in LMFDB) 5.3.f_r_bx_eh_hw $6$ (not in LMFDB) 5.3.h_bd_dl_if_ps $6$ (not in LMFDB) 5.3.ae_f_b_ay_co $12$ (not in LMFDB) 5.3.ac_ab_ab_ag_bq $12$ (not in LMFDB) 5.3.c_ab_b_ag_abq $12$ (not in LMFDB) 5.3.e_f_ab_ay_aco $12$ (not in LMFDB) 5.3.ae_l_ax_bq_acu $24$ (not in LMFDB) 5.3.ac_f_an_y_abk $24$ (not in LMFDB) 5.3.c_f_n_y_bk $24$ (not in LMFDB) 5.3.e_l_x_bq_cu $24$ (not in LMFDB)