# Properties

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

## Invariants

 Base field: $\F_{3^{4}}$ Dimension: $2$ L-polynomial: $( 1 - 17 x + 81 x^{2} )( 1 - 16 x + 81 x^{2} )$ Frobenius angles: $\pm0.106600758076$, $\pm0.151478024726$ Angle rank: $2$ (numerical) Jacobians: 0

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 is principally polarizable, but does not contain a Jacobian.

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 4290 41621580 281904634440 1853075318814720 12158005941111137250 79766858524967437824000 523348000830632514498085410 3433684088563591517712280903680 22528399708717801570887258860748360 147808829493376537074427048595644759500

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 49 6341 530452 43048001 3486882049 282431007458 22876808532529 1853020333626881 150094636388168212 12157665465557404901

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{4}}$
 The isogeny class factors as 1.81.ar $\times$ 1.81.aq 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.
 Twist Extension Degree Common base change 2.81.ab_aeg $2$ (not in LMFDB) 2.81.b_aeg $2$ (not in LMFDB) 2.81.bh_qs $2$ (not in LMFDB)