# Properties

 Label 5.3.ak_bx_agd_ox_abct 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: $( 1 - 3 x + 3 x^{2} )^{2}( 1 - 4 x + 10 x^{2} - 21 x^{3} + 30 x^{4} - 36 x^{5} + 27 x^{6} )$ Frobenius angles: $\pm0.0145064862012$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.383559653096$, $\pm0.564732805964$ 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 $A(\F_{q^r})$ 7 44247 11398576 2968663971 907636535197 217880406261504 50713356559716571 12492634994784032907 2962941353177669635072 699162416223096543672927

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -6 8 21 68 264 773 2220 6740 19740 57488

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ad 2 $\times$ 3.3.ae_k_av 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_av : 6.0.309123.1.
Endomorphism algebra over $\overline{\F}_{3}$
 The base change of $A$ to $\F_{3^{6}}$ is 1.729.cc 2 $\times$ 3.729.acn_djq_adrer. 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.acn_djq_adrer : 6.0.309123.1.
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_ai_acx. The endomorphism algebra for each factor is: 1.9.ad 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$ 3.9.e_ai_acx : 6.0.309123.1.
• Endomorphism algebra over $\F_{3^{3}}$  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a 2 $\times$ 3.27.ah_ai_hh. The endomorphism algebra for each factor is: 1.27.a 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$ 3.27.ah_ai_hh : 6.0.309123.1.

## 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_d_aj_bb $2$ (not in LMFDB) 5.3.e_h_j_j_j $2$ (not in LMFDB) 5.3.k_bx_gd_ox_bct $2$ (not in LMFDB) 5.3.ah_bc_adg_hq_aoo $3$ (not in LMFDB) 5.3.ae_b_p_aj_abb $3$ (not in LMFDB) 5.3.ae_h_aj_j_aj $3$ (not in LMFDB) 5.3.ae_h_aj_bb_acl $3$ (not in LMFDB) 5.3.ae_q_abt_dv_ahq $3$ (not in LMFDB) 5.3.ab_ac_a_a_s $3$ (not in LMFDB) 5.3.ab_e_ag_a_as $3$ (not in LMFDB) 5.3.ab_e_ag_s_as $3$ (not in LMFDB) 5.3.c_af_ap_j_cl $3$ (not in LMFDB) 5.3.c_b_ad_aj_abb $3$ (not in LMFDB) 5.3.c_b_ad_j_bb $3$ (not in LMFDB) 5.3.c_e_d_aj_as $3$ (not in LMFDB) 5.3.c_k_p_bt_cc $3$ (not in LMFDB) 5.3.f_k_g_as_acc $3$ (not in LMFDB) 5.3.f_q_bk_cu_ew $3$ (not in LMFDB) 5.3.i_z_bb_abt_agp $3$ (not in LMFDB) 5.3.i_bf_cx_ff_ir $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 5.3.ac_b_d_aj_bb $2$ (not in LMFDB) 5.3.e_h_j_j_j $2$ (not in LMFDB) 5.3.k_bx_gd_ox_bct $2$ (not in LMFDB) 5.3.ah_bc_adg_hq_aoo $3$ (not in LMFDB) 5.3.ae_b_p_aj_abb $3$ (not in LMFDB) 5.3.ae_h_aj_j_aj $3$ (not in LMFDB) 5.3.ae_h_aj_bb_acl $3$ (not in LMFDB) 5.3.ae_q_abt_dv_ahq $3$ (not in LMFDB) 5.3.ab_ac_a_a_s $3$ (not in LMFDB) 5.3.ab_e_ag_a_as $3$ (not in LMFDB) 5.3.ab_e_ag_s_as $3$ (not in LMFDB) 5.3.c_af_ap_j_cl $3$ (not in LMFDB) 5.3.c_b_ad_aj_abb $3$ (not in LMFDB) 5.3.c_b_ad_j_bb $3$ (not in LMFDB) 5.3.c_e_d_aj_as $3$ (not in LMFDB) 5.3.c_k_p_bt_cc $3$ (not in LMFDB) 5.3.f_k_g_as_acc $3$ (not in LMFDB) 5.3.f_q_bk_cu_ew $3$ (not in LMFDB) 5.3.i_z_bb_abt_agp $3$ (not in LMFDB) 5.3.i_bf_cx_ff_ir $3$ (not in LMFDB) 5.3.ae_n_abh_cr_aff $4$ (not in LMFDB) 5.3.e_n_bh_cr_ff $4$ (not in LMFDB) 5.3.ai_z_abb_abt_gp $6$ (not in LMFDB) 5.3.ai_bf_acx_ff_air $6$ (not in LMFDB) 5.3.af_k_ag_as_cc $6$ (not in LMFDB) 5.3.af_q_abk_cu_aew $6$ (not in LMFDB) 5.3.ac_af_p_j_acl $6$ (not in LMFDB) 5.3.ac_b_d_j_abb $6$ (not in LMFDB) 5.3.ac_e_ad_aj_s $6$ (not in LMFDB) 5.3.ac_k_ap_bt_acc $6$ (not in LMFDB) 5.3.b_ac_a_a_as $6$ (not in LMFDB) 5.3.b_e_g_a_s $6$ (not in LMFDB) 5.3.b_e_g_s_s $6$ (not in LMFDB) 5.3.c_af_ap_j_cl $6$ (not in LMFDB) 5.3.e_b_ap_aj_bb $6$ (not in LMFDB) 5.3.e_h_j_bb_cl $6$ (not in LMFDB) 5.3.e_q_bt_dv_hq $6$ (not in LMFDB) 5.3.h_bc_dg_hq_oo $6$ (not in LMFDB) 5.3.ae_e_d_av_cc $12$ (not in LMFDB) 5.3.ac_ai_v_p_adm $12$ (not in LMFDB) 5.3.ac_ac_j_ad_as $12$ (not in LMFDB) 5.3.ac_b_d_ad_aj $12$ (not in LMFDB) 5.3.ac_h_aj_bh_abt $12$ (not in LMFDB) 5.3.c_ai_av_p_dm $12$ (not in LMFDB) 5.3.c_ac_aj_ad_s $12$ (not in LMFDB) 5.3.c_b_ad_ad_j $12$ (not in LMFDB) 5.3.c_h_j_bh_bt $12$ (not in LMFDB) 5.3.e_e_ad_av_acc $12$ (not in LMFDB) 5.3.ae_k_av_bn_acu $24$ (not in LMFDB) 5.3.ac_ac_j_d_abk $24$ (not in LMFDB) 5.3.ac_e_ad_v_abk $24$ (not in LMFDB) 5.3.c_ac_aj_d_bk $24$ (not in LMFDB) 5.3.c_e_d_v_bk $24$ (not in LMFDB) 5.3.e_k_v_bn_cu $24$ (not in LMFDB)