Properties

Label 2.81.abg_py
Base Field $\F_{3^{4}}$
Dimension $2$
Ordinary No
$p$-rank $1$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{3^{4}}$
Dimension:  $2$
L-polynomial:  $( 1 - 9 x )^{2}( 1 - 14 x + 81 x^{2} )$
Frobenius angles:  $0$, $0$, $\pm0.216346895939$
Angle rank:  $1$ (numerical)
Jacobians:  3

This isogeny class is not simple.

Newton polygon

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

Point counts

This isogeny class contains the Jacobians of 3 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 4352 41779200 282004486400 1852970355916800 12157651951711430912 79766320794946007040000 523347404273394977385128192 3433683554967536294019937075200 22528399313941250143604756860985600 147808829255717821926156950744494080000

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 50 6366 530642 43045566 3486780530 282429103518 22876782455570 1853020045666686 150094633757983922 12157665446009349726

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{4}}$
The isogeny class factors as 1.81.as $\times$ 1.81.ao and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
All geometric endomorphisms are defined over $\F_{3^{4}}$.

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon base change
2.81.ae_adm$2$(not in LMFDB)
2.81.e_adm$2$(not in LMFDB)
2.81.bg_py$2$(not in LMFDB)
2.81.af_bk$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.81.ae_adm$2$(not in LMFDB)
2.81.e_adm$2$(not in LMFDB)
2.81.bg_py$2$(not in LMFDB)
2.81.af_bk$3$(not in LMFDB)
2.81.ao_gg$4$(not in LMFDB)
2.81.o_gg$4$(not in LMFDB)
2.81.ax_lc$6$(not in LMFDB)
2.81.f_bk$6$(not in LMFDB)
2.81.x_lc$6$(not in LMFDB)