# Properties

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

## Invariants

 Base field: $\F_{3^{4}}$ Dimension: $2$ L-polynomial: $( 1 - 9 x )^{2}( 1 - 13 x + 81 x^{2} )$ Frobenius angles: $0$, $0$, $\pm0.243120792737$ 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.

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 4416 41952000 282165601536 1853017994112000 12157574551206148416 79766181704571328512000 523347284535421141700840256 3433683503605204513284828672000 22528399332059567325095452566195456 147808829311383034760430255547020000000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 51 6393 530946 43046673 3486758331 282428611038 22876777221531 1853020017948513 150094633878696546 12157665450587960073

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{4}}$
 The isogeny class factors as 1.81.as $\times$ 1.81.an and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: 1.81.as : the quaternion algebra over $$\Q$$ ramified at $3$ and $\infty$. 1.81.an : $$\Q(\sqrt{-155})$$.
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.
 Twist Extension Degree Common base change 2.81.af_acu $2$ (not in LMFDB) 2.81.f_acu $2$ (not in LMFDB) 2.81.bf_pg $2$ (not in LMFDB) 2.81.ae_bt $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.81.af_acu $2$ (not in LMFDB) 2.81.f_acu $2$ (not in LMFDB) 2.81.bf_pg $2$ (not in LMFDB) 2.81.ae_bt $3$ (not in LMFDB) 2.81.an_gg $4$ (not in LMFDB) 2.81.n_gg $4$ (not in LMFDB) 2.81.aw_kt $6$ (not in LMFDB) 2.81.e_bt $6$ (not in LMFDB) 2.81.w_kt $6$ (not in LMFDB)