# Properties

 Label 2.81.abd_nw 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: $( 1 - 9 x )^{2}( 1 - 11 x + 81 x^{2} )$ Frobenius angles: $0$, $0$, $\pm0.290722850198$ 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 $A(\F_{q^r})$ 4544 42259200 282366995456 1852947763276800 12157313476606637504 79765934432794122240000 523347196677639745627726784 3433683577738698105284518195200 22528399470262777315566881461185536 147808829410770497330604832243834080000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 53 6441 531326 43045041 3486683453 282427735518 22876773381053 1853020057955361 150094634799470366 12157665458762840601

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{4}}$
 The isogeny class factors as 1.81.as $\times$ 1.81.al 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.al : $$\Q(\sqrt{-203})$$.
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.ah_abk $2$ (not in LMFDB) 2.81.h_abk $2$ (not in LMFDB) 2.81.bd_nw $2$ (not in LMFDB) 2.81.ac_cl $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.81.ah_abk $2$ (not in LMFDB) 2.81.h_abk $2$ (not in LMFDB) 2.81.bd_nw $2$ (not in LMFDB) 2.81.ac_cl $3$ (not in LMFDB) 2.81.al_gg $4$ (not in LMFDB) 2.81.l_gg $4$ (not in LMFDB) 2.81.au_kb $6$ (not in LMFDB) 2.81.c_cl $6$ (not in LMFDB) 2.81.u_kb $6$ (not in LMFDB)