Properties

Label 5.3.ak_bv_afj_lr_avj
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:  $( 1 - 3 x + 3 x^{2} )^{2}( 1 - 4 x + 8 x^{2} - 13 x^{3} + 24 x^{4} - 36 x^{5} + 27 x^{6} )$
Frobenius angles:  $\pm0.102762435325$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.278353759721$, $\pm0.643265352440$
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 38759 11464432 5430174659 1229650350257 220397560264448 53076707805199099 12768830646189408427 2963011331031393467392 721245798565095685030919

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -6 4 21 116 334 781 2318 6884 19740 59324

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
The isogeny class factors as 1.3.ad 2 $\times$ 3.3.ae_i_an 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.acf_dba_adgnl. 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_f_ab_j_abh$2$(not in LMFDB)
5.3.ac_ab_h_d_av$2$(not in LMFDB)
5.3.c_ab_ah_d_v$2$(not in LMFDB)
5.3.e_f_b_j_bh$2$(not in LMFDB)
5.3.k_bv_fj_lr_vj$2$(not in LMFDB)
5.3.ah_ba_acs_ga_ali$3$(not in LMFDB)
5.3.ae_o_abl_dd_afu$3$(not in LMFDB)
5.3.ab_c_ae_g_ag$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
5.3.ae_f_ab_j_abh$2$(not in LMFDB)
5.3.ac_ab_h_d_av$2$(not in LMFDB)
5.3.c_ab_ah_d_v$2$(not in LMFDB)
5.3.e_f_b_j_bh$2$(not in LMFDB)
5.3.k_bv_fj_lr_vj$2$(not in LMFDB)
5.3.ah_ba_acs_ga_ali$3$(not in LMFDB)
5.3.ae_o_abl_dd_afu$3$(not in LMFDB)
5.3.ab_c_ae_g_ag$3$(not in LMFDB)
5.3.ae_l_az_cf_aeh$4$(not in LMFDB)
5.3.e_l_z_cf_eh$4$(not in LMFDB)
5.3.ac_ab_h_d_av$6$(not in LMFDB)
5.3.b_c_e_g_g$6$(not in LMFDB)
5.3.e_o_bl_dd_fu$6$(not in LMFDB)
5.3.h_ba_cs_ga_li$6$(not in LMFDB)
5.3.ae_c_l_ap_g$12$(not in LMFDB)
5.3.e_c_al_ap_ag$12$(not in LMFDB)
5.3.ae_i_an_bh_acu$24$(not in LMFDB)
5.3.e_i_n_bh_cu$24$(not in LMFDB)