Properties

Label 5.3.ak_bw_aft_nh_azb
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:  $( 1 - 3 x + 3 x^{2} )^{2}( 1 - 4 x + 9 x^{2} - 17 x^{3} + 27 x^{4} - 36 x^{5} + 27 x^{6} )$
Frobenius angles:  $\pm0.0653366913680$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.328985474983$, $\pm0.609104440316$
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
$A(\F_{q^r})$ 7 41503 11217472 4159306151 1059191953232 208570315706368 50778450716647997 12902146522011609863 3010965622576573221952 712713029971932025127168

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -6 6 21 94 299 741 2220 6950 20055 58621

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
The isogeny class factors as 1.3.ad 2 $\times$ 3.3.ae_j_ar 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.adt_hiy_ajlhg. 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_g_af_j_av$2$(not in LMFDB)
5.3.ac_a_f_ad_d$2$(not in LMFDB)
5.3.c_a_af_ad_ad$2$(not in LMFDB)
5.3.e_g_f_j_v$2$(not in LMFDB)
5.3.k_bw_ft_nh_zb$2$(not in LMFDB)
5.3.ah_bb_acz_gv_amy$3$(not in LMFDB)
5.3.ae_g_af_j_av$3$(not in LMFDB)
5.3.ae_p_abp_dm_ags$3$(not in LMFDB)
5.3.ab_d_af_d_am$3$(not in LMFDB)
5.3.c_a_af_ad_ad$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
5.3.ae_g_af_j_av$2$(not in LMFDB)
5.3.ac_a_f_ad_d$2$(not in LMFDB)
5.3.c_a_af_ad_ad$2$(not in LMFDB)
5.3.e_g_f_j_v$2$(not in LMFDB)
5.3.k_bw_ft_nh_zb$2$(not in LMFDB)
5.3.ah_bb_acz_gv_amy$3$(not in LMFDB)
5.3.ae_g_af_j_av$3$(not in LMFDB)
5.3.ae_p_abp_dm_ags$3$(not in LMFDB)
5.3.ab_d_af_d_am$3$(not in LMFDB)
5.3.c_a_af_ad_ad$3$(not in LMFDB)
5.3.ae_m_abd_cl_aet$4$(not in LMFDB)
5.3.e_m_bd_cl_et$4$(not in LMFDB)
5.3.b_d_f_d_m$6$(not in LMFDB)
5.3.e_p_bp_dm_gs$6$(not in LMFDB)
5.3.h_bb_cz_gv_my$6$(not in LMFDB)
5.3.ae_d_h_as_be$12$(not in LMFDB)
5.3.e_d_ah_as_abe$12$(not in LMFDB)
5.3.ae_j_ar_bk_acu$24$(not in LMFDB)
5.3.e_j_r_bk_cu$24$(not in LMFDB)