# Properties

 Label 5.3.ak_bs_aeh_hh_all Base Field $\F_{3}$ Dimension $5$ Ordinary No $p$-rank $2$ Principally polarizable Yes Contains a Jacobian No

## Invariants

 Base field: $\F_{3}$ Dimension: $5$ L-polynomial: $( 1 - 3 x + 3 x^{2} )^{3}( 1 - x - x^{2} - 3 x^{3} + 9 x^{4} )$ Frobenius angles: $\pm0.126866938441$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.718153680921$ Angle rank: $2$ (numerical)

This isogeny class is not simple.

## Newton polygon

 $p$-rank: $2$ Slopes: $[0, 0, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1, 1]$

## 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 22295 9988160 6122764375 1202111664400 259384684482560 60645830392084495 12711983771875854375 2988178715703504464960 717949120226839108025600

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -6 -2 15 126 329 901 2612 6854 19905 59053

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ad 3 $\times$ 2.3.ab_ab 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 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-3})$$$)$ 2.3.ab_ab : 4.0.10933.1.
Endomorphism algebra over $\overline{\F}_{3}$
 The base change of $A$ to $\F_{3^{6}}$ is 1.729.cc 3 $\times$ 2.729.j_jt. The endomorphism algebra for each factor is: 1.729.cc 3 : $\mathrm{M}_{3}(B)$, where $B$ is the quaternion algebra over $$\Q$$ ramified at $3$ and $\infty$. 2.729.j_jt : 4.0.10933.1.
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 3 $\times$ 2.9.ad_n. The endomorphism algebra for each factor is: 1.9.ad 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-3})$$$)$ 2.9.ad_n : 4.0.10933.1.
• Endomorphism algebra over $\F_{3^{3}}$  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a 3 $\times$ 2.27.an_dl. The endomorphism algebra for each factor is: 1.27.a 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-3})$$$)$ 2.27.an_dl : 4.0.10933.1.

## 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_ba_abh_abb_ff $2$ (not in LMFDB) 5.3.ac_ae_p_j_acl $2$ (not in LMFDB) 5.3.e_c_aj_j_cl $2$ (not in LMFDB) 5.3.k_bs_eh_hh_ll $2$ (not in LMFDB) 5.3.ah_x_abz_dv_agy $3$ (not in LMFDB) 5.3.ae_c_j_j_acl $3$ (not in LMFDB) 5.3.ae_l_abb_cc_adm $3$ (not in LMFDB) 5.3.ab_ab_ad_j_a $3$ (not in LMFDB) 5.3.ab_i_am_bb_acc $3$ (not in LMFDB) 5.3.c_ae_ap_j_cl $3$ (not in LMFDB) 5.3.c_f_d_a_as $3$ (not in LMFDB) 5.3.f_l_j_aj_abk $3$ (not in LMFDB) 5.3.i_ba_bh_abb_aff $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 5.3.ai_ba_abh_abb_ff $2$ (not in LMFDB) 5.3.ac_ae_p_j_acl $2$ (not in LMFDB) 5.3.e_c_aj_j_cl $2$ (not in LMFDB) 5.3.k_bs_eh_hh_ll $2$ (not in LMFDB) 5.3.ah_x_abz_dv_agy $3$ (not in LMFDB) 5.3.ae_c_j_j_acl $3$ (not in LMFDB) 5.3.ae_l_abb_cc_adm $3$ (not in LMFDB) 5.3.ab_ab_ad_j_a $3$ (not in LMFDB) 5.3.ab_i_am_bb_acc $3$ (not in LMFDB) 5.3.c_ae_ap_j_cl $3$ (not in LMFDB) 5.3.c_f_d_a_as $3$ (not in LMFDB) 5.3.f_l_j_aj_abk $3$ (not in LMFDB) 5.3.i_ba_bh_abb_aff $3$ (not in LMFDB) 5.3.ae_i_ap_bn_add $4$ (not in LMFDB) 5.3.ac_c_d_d_aj $4$ (not in LMFDB) 5.3.c_c_ad_d_j $4$ (not in LMFDB) 5.3.e_i_p_bn_dd $4$ (not in LMFDB) 5.3.af_l_aj_aj_bk $6$ (not in LMFDB) 5.3.ac_f_ad_a_s $6$ (not in LMFDB) 5.3.b_ab_d_j_a $6$ (not in LMFDB) 5.3.b_i_m_bb_cc $6$ (not in LMFDB) 5.3.c_f_d_a_as $6$ (not in LMFDB) 5.3.e_l_bb_cc_dm $6$ (not in LMFDB) 5.3.h_x_bz_dv_gy $6$ (not in LMFDB) 5.3.ab_ab_am_s_j $9$ (not in LMFDB) 5.3.ab_ab_g_a_aj $9$ (not in LMFDB) 5.3.ae_ab_v_ag_acc $12$ (not in LMFDB) 5.3.ac_ah_v_m_adm $12$ (not in LMFDB) 5.3.ab_ae_a_d_s $12$ (not in LMFDB) 5.3.ab_f_aj_v_abk $12$ (not in LMFDB) 5.3.b_ae_a_d_as $12$ (not in LMFDB) 5.3.b_f_j_v_bk $12$ (not in LMFDB) 5.3.c_ah_av_m_dm $12$ (not in LMFDB) 5.3.e_ab_av_ag_cc $12$ (not in LMFDB) 5.3.b_ab_ag_a_j $18$ (not in LMFDB) 5.3.b_ab_m_s_aj $18$ (not in LMFDB) 5.3.ae_f_ad_y_acu $24$ (not in LMFDB) 5.3.ac_ab_j_g_abk $24$ (not in LMFDB) 5.3.ab_c_ag_p_as $24$ (not in LMFDB) 5.3.b_c_g_p_s $24$ (not in LMFDB) 5.3.c_ab_aj_g_bk $24$ (not in LMFDB) 5.3.e_f_d_y_cu $24$ (not in LMFDB)