Properties

Label 2.81.abf_pk
Base Field $\F_{3^{4}}$
Dimension $2$
Ordinary Yes
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian Yes

Learn more about

Invariants

Base field:  $\F_{3^{4}}$
Dimension:  $2$
L-polynomial:  $( 1 - 17 x + 81 x^{2} )( 1 - 14 x + 81 x^{2} )$
Frobenius angles:  $\pm0.106600758076$, $\pm0.216346895939$
Angle rank:  $2$ (numerical)
Jacobians:  14

This isogeny class is not simple.

Newton polygon

This isogeny class is ordinary.

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

Point counts

This isogeny class contains the Jacobians of 14 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 4420 42007680 282364186000 1853405894545920 12158106367827530500 79766748462001176576000 523347775997496339979781380 3433683857278199941704279214080 22528399545595777515915934575706000 147808829423466078703027491480481872000

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 51 6401 531318 43055681 3486910851 282430617758 22876798704531 1853020208811521 150094635301373718 12157665459807085601

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{4}}$
The isogeny class factors as 1.81.ar $\times$ 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 is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.81.ad_acy$2$(not in LMFDB)
2.81.d_acy$2$(not in LMFDB)
2.81.bf_pk$2$(not in LMFDB)