# Properties

 Label 1.81.as Base Field $\F_{3^{4}}$ Dimension $1$ Ordinary No $p$-rank $0$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{3^{4}}$ Dimension: $1$ L-polynomial: $( 1 - 9 x )^{2}$ Frobenius angles: $0$, $0$ Angle rank: $0$ (numerical) Number field: $$\Q$$ Galois group: Trivial Jacobians: 1

This isogeny class is simple and geometrically simple.

## Newton polygon

This isogeny class is supersingular.

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

## Point counts

This isogeny class contains the Jacobians of 1 curves, and hence is principally polarizable:

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 64 6400 529984 43033600 3486666304 282428473600 22876782889024 1853020102758400 150094634522158144 12157665452083360000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 64 6400 529984 43033600 3486666304 282428473600 22876782889024 1853020102758400 150094634522158144 12157665452083360000

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{4}}$
 The endomorphism algebra of this simple isogeny class is the quaternion algebra over $$\Q$$ ramified at $3$ and $\infty$.
All geometric endomorphisms are defined over $\F_{3^{4}}$.

## Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{3^{4}}$.

 Subfield Primitive Model $\F_{3}$ 1.3.a

## Twists

Below are some of the twists of this isogeny class.
 Twist Extension Degree Common base change 1.81.s $2$ (not in LMFDB) 1.81.j $3$ (not in LMFDB) 1.81.a $4$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 1.81.s $2$ (not in LMFDB) 1.81.j $3$ (not in LMFDB) 1.81.a $4$ (not in LMFDB) 1.81.aj $6$ (not in LMFDB)