Properties

Label 5.3.ak_bt_aer_ix_apd
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:  $( 1 - 3 x + 3 x^{2} )^{2}( 1 - 4 x + 6 x^{2} - 7 x^{3} + 18 x^{4} - 36 x^{5} + 27 x^{6} )$
Frobenius angles:  $\pm0.0452398905210$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.239335307006$, $\pm0.691360448188$
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
$A(\F_{q^r})$ 5 24255 8988560 5267337075 1082283492775 215575726544640 53420925605073745 12017202035056893675 2874085776613431618560 714887926089295297652775

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -6 0 15 116 304 765 2332 6484 19140 58800

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
The isogeny class factors as 1.3.ad 2 $\times$ 3.3.ae_g_ah 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.cc 2 $\times$ 3.729.acv_eli_afacr. 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.ae_d_f_j_abz$2$(not in LMFDB)
5.3.ac_ad_n_d_abn$2$(not in LMFDB)
5.3.c_ad_an_d_bn$2$(not in LMFDB)
5.3.e_d_af_j_bz$2$(not in LMFDB)
5.3.k_bt_er_ix_pd$2$(not in LMFDB)
5.3.ah_y_acg_eq_aio$3$(not in LMFDB)
5.3.ae_d_f_j_abz$3$(not in LMFDB)
5.3.ae_m_abf_cl_aek$3$(not in LMFDB)
5.3.ab_a_ae_g_ag$3$(not in LMFDB)
5.3.c_ad_an_d_bn$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
5.3.ae_d_f_j_abz$2$(not in LMFDB)
5.3.ac_ad_n_d_abn$2$(not in LMFDB)
5.3.c_ad_an_d_bn$2$(not in LMFDB)
5.3.e_d_af_j_bz$2$(not in LMFDB)
5.3.k_bt_er_ix_pd$2$(not in LMFDB)
5.3.ah_y_acg_eq_aio$3$(not in LMFDB)
5.3.ae_d_f_j_abz$3$(not in LMFDB)
5.3.ae_m_abf_cl_aek$3$(not in LMFDB)
5.3.ab_a_ae_g_ag$3$(not in LMFDB)
5.3.c_ad_an_d_bn$3$(not in LMFDB)
5.3.ae_j_at_bt_adp$4$(not in LMFDB)
5.3.e_j_t_bt_dp$4$(not in LMFDB)
5.3.ah_y_acg_eq_aio$6$(not in LMFDB)
5.3.ae_d_f_j_abz$6$(not in LMFDB)
5.3.ae_m_abf_cl_aek$6$(not in LMFDB)
5.3.ac_ad_n_d_abn$6$(not in LMFDB)
5.3.ab_a_ae_g_ag$6$(not in LMFDB)
5.3.b_a_e_g_g$6$(not in LMFDB)
5.3.e_d_af_j_bz$6$(not in LMFDB)
5.3.e_m_bf_cl_ek$6$(not in LMFDB)
5.3.h_y_cg_eq_io$6$(not in LMFDB)
5.3.ae_a_r_aj_abe$12$(not in LMFDB)
5.3.ae_j_at_bt_adp$12$(not in LMFDB)
5.3.e_a_ar_aj_be$12$(not in LMFDB)
5.3.e_j_t_bt_dp$12$(not in LMFDB)
5.3.ae_g_ah_bb_acu$24$(not in LMFDB)
5.3.e_g_h_bb_cu$24$(not in LMFDB)