Properties

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

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Invariants

Base field:  $\F_{3^{4}}$
Dimension:  $2$
L-polynomial:  $( 1 - 9 x )^{2}( 1 - 11 x + 81 x^{2} )$
Frobenius angles:  $0$, $0$, $\pm0.290722850198$
Angle rank:  $1$ (numerical)
Jacobians:  0

This isogeny class is not simple.

Newton polygon

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

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
$A(\F_{q^r})$ 4544 42259200 282366995456 1852947763276800 12157313476606637504 79765934432794122240000 523347196677639745627726784 3433683577738698105284518195200 22528399470262777315566881461185536 147808829410770497330604832243834080000

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 53 6441 531326 43045041 3486683453 282427735518 22876773381053 1853020057955361 150094634799470366 12157665458762840601

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{4}}$
The isogeny class factors as 1.81.as $\times$ 1.81.al 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.ah_abk$2$(not in LMFDB)
2.81.h_abk$2$(not in LMFDB)
2.81.bd_nw$2$(not in LMFDB)
2.81.ac_cl$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.81.ah_abk$2$(not in LMFDB)
2.81.h_abk$2$(not in LMFDB)
2.81.bd_nw$2$(not in LMFDB)
2.81.ac_cl$3$(not in LMFDB)
2.81.al_gg$4$(not in LMFDB)
2.81.l_gg$4$(not in LMFDB)
2.81.au_kb$6$(not in LMFDB)
2.81.c_cl$6$(not in LMFDB)
2.81.u_kb$6$(not in LMFDB)