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

Learn more about

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.

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 9 59535 15494976 3068731575 976869879984 255838168535040 55864604080981719 12681258131107995975 3003469397327024990784 715886088626675908972800

Point counts of the (virtual) curve

$r$ 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:
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:
All geometric endomorphisms are defined over $\F_{3^{6}}$.
Remainder of endomorphism lattice by field

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon 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.
TwistExtension DegreeCommon 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)