# Properties

 Label 2.81.abk_ss Base Field $\F_{3^{4}}$ Dimension $2$ Ordinary No $p$-rank $0$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

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

This isogeny class is not simple.

## Newton polygon

This isogeny class is supersingular.

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

## Point counts

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

• $y^2=ax^5+2a$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 4096 40960000 280883040256 1851890728960000 12156841915449020416 79765842700025896960000 523347195351541271899672576 3433683501226751295288770560000 22528399312340227300247719345524736 147808829244781290288164388889600000000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 46 6238 528526 43020478 3486548206 282427410718 22876773323086 1853020016664958 150094633747317166 12157665445109791198

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{4}}$
 The isogeny class factors as 1.81.as 2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ 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 $\F_{3}$ 2.3.a_ag $\F_{3}$ 2.3.a_g

## Twists

Below are some of the twists of this isogeny class.
 Twist Extension Degree Common base change 2.81.a_agg $2$ (not in LMFDB) 2.81.bk_ss $2$ (not in LMFDB) 2.81.aj_a $3$ (not in LMFDB) 2.81.s_jj $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.81.a_agg $2$ (not in LMFDB) 2.81.bk_ss $2$ (not in LMFDB) 2.81.aj_a $3$ (not in LMFDB) 2.81.s_jj $3$ (not in LMFDB) 2.81.as_gg $4$ (not in LMFDB) 2.81.a_gg $4$ (not in LMFDB) 2.81.s_gg $4$ (not in LMFDB) 2.81.j_dd $5$ (not in LMFDB) 2.81.abb_mm $6$ (not in LMFDB) 2.81.as_jj $6$ (not in LMFDB) 2.81.a_dd $6$ (not in LMFDB) 2.81.j_a $6$ (not in LMFDB) 2.81.bb_mm $6$ (not in LMFDB) 2.81.a_a $8$ (not in LMFDB) 2.81.aj_dd $10$ (not in LMFDB) 2.81.aj_gg $12$ (not in LMFDB) 2.81.a_add $12$ (not in LMFDB) 2.81.j_gg $12$ (not in LMFDB)