# Properties

 Label 2.81.abj_sa 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: $( 1 - 9 x )^{2}( 1 - 17 x + 81 x^{2} )$ Frobenius angles: $0$, $0$, $\pm0.106600758076$ 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 $A(\F_{q^r})$ 4160 41184000 281241309440 1852326013824000 12157296301288424000 79766270364517761024000 523347567075494241117976640 3433683803537410211490983424000 22528399543994754656095934693423360 147808829412529547052623058652140000000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 47 6273 529202 43030593 3486678527 282428924958 22876789572047 1853020179809793 150094635290706962 12157665458907527073

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{4}}$
 The isogeny class factors as 1.81.as $\times$ 1.81.ar 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.ar : $$\Q(\sqrt{-35})$$.
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.ab_afo $2$ (not in LMFDB) 2.81.b_afo $2$ (not in LMFDB) 2.81.bj_sa $2$ (not in LMFDB) 2.81.ai_j $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.81.ab_afo $2$ (not in LMFDB) 2.81.b_afo $2$ (not in LMFDB) 2.81.bj_sa $2$ (not in LMFDB) 2.81.ai_j $3$ (not in LMFDB) 2.81.ar_gg $4$ (not in LMFDB) 2.81.r_gg $4$ (not in LMFDB) 2.81.aba_md $6$ (not in LMFDB) 2.81.i_j $6$ (not in LMFDB) 2.81.ba_md $6$ (not in LMFDB)