Properties

Label 5.3.ak_by_agk_pu_abep
Base Field $\F_{3}$
Dimension $5$
Ordinary No
$p$-rank $4$
Principally polarizable Yes
Contains a Jacobian No

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Invariants

Base field:  $\F_{3}$
Dimension:  $5$
L-polynomial:  $( 1 - 3 x + 3 x^{2} )( 1 - 7 x + 26 x^{2} - 67 x^{3} + 131 x^{4} - 201 x^{5} + 234 x^{6} - 189 x^{7} + 81 x^{8} )$
Frobenius angles:  $\pm0.0365491909691$, $\pm0.166666666667$, $\pm0.234353293111$, $\pm0.355928212636$, $\pm0.548250857189$
Angle rank:  $4$ (numerical)

This isogeny class is not simple.

Newton polygon

$p$-rank:  $4$
Slopes:  $[0, 0, 0, 0, 1/2, 1/2, 1, 1, 1, 1]$

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 59031 15595524 3463289739 933273474489 209082889676304 47635494725780901 12010765592522223891 2968078619705306842476 709061345815096794379671

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -6 10 30 82 269 742 2080 6482 19776 58315

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
The isogeny class factors as 1.3.ad $\times$ 4.3.ah_ba_acp_fb 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 $\times$ 4.729.abq_boh_acfrw_djcin. 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_i_ak_i_aj$2$(not in LMFDB)
5.3.e_i_k_i_j$2$(not in LMFDB)
5.3.k_by_gk_pu_bep$2$(not in LMFDB)
5.3.ah_bd_adk_ib_apm$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
5.3.ae_i_ak_i_aj$2$(not in LMFDB)
5.3.e_i_k_i_j$2$(not in LMFDB)
5.3.k_by_gk_pu_bep$2$(not in LMFDB)
5.3.ah_bd_adk_ib_apm$3$(not in LMFDB)
5.3.e_i_k_i_j$6$(not in LMFDB)
5.3.h_bd_dk_ib_pm$6$(not in LMFDB)