# Properties

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

## Invariants

 Base field: $\F_{3^{4}}$ Dimension: $2$ L-polynomial: $( 1 - 9 x )^{2}( 1 - 9 x + 81 x^{2} )$ Frobenius angles: $0$, $0$, $\pm0.333333333333$ Angle rank: $0$ (numerical) Jacobians: 0

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 is principally polarizable, but does not contain a Jacobian.

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 4672 42515200 282428473600 1852737759308800 12157047799607605312 79765842700025896960000 523347304770484649831904832 3433683740526069407785296179200 22528399544939174111650877280774400 147808829371954765040867006688602080000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 55 6481 531442 43040161 3486607255 282427410718 22876778106055 1853020145805121 150094635296999122 12157665455570144401

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{4}}$
 The isogeny class factors as 1.81.as $\times$ 1.81.aj 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.aj : $$\Q(\sqrt{-3})$$.
Endomorphism algebra over $\overline{\F}_{3^{4}}$
 The base change of $A$ to $\F_{3^{24}}$ is 1.282429536481.acimic 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 $\F_{3^{24}}$.
Remainder of endomorphism lattice by field
• Endomorphism algebra over $\F_{3^{8}}$  The base change of $A$ to $\F_{3^{8}}$ is 1.6561.agg $\times$ 1.6561.dd. The endomorphism algebra for each factor is: 1.6561.agg : the quaternion algebra over $$\Q$$ ramified at $3$ and $\infty$. 1.6561.dd : $$\Q(\sqrt{-3})$$.
• Endomorphism algebra over $\F_{3^{12}}$  The base change of $A$ to $\F_{3^{12}}$ is 1.531441.acec $\times$ 1.531441.cec. The endomorphism algebra for each factor is: 1.531441.acec : the quaternion algebra over $$\Q$$ ramified at $3$ and $\infty$. 1.531441.cec : the quaternion algebra over $$\Q$$ ramified at $3$ and $\infty$.

## 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.aj_a $2$ (not in LMFDB) 2.81.j_a $2$ (not in LMFDB) 2.81.bb_mm $2$ (not in LMFDB) 2.81.a_agg $3$ (not in LMFDB) 2.81.a_dd $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.81.aj_a $2$ (not in LMFDB) 2.81.j_a $2$ (not in LMFDB) 2.81.bb_mm $2$ (not in LMFDB) 2.81.a_agg $3$ (not in LMFDB) 2.81.a_dd $3$ (not in LMFDB) 2.81.aj_gg $4$ (not in LMFDB) 2.81.j_gg $4$ (not in LMFDB) 2.81.abk_ss $6$ (not in LMFDB) 2.81.as_jj $6$ (not in LMFDB) 2.81.aj_a $6$ (not in LMFDB) 2.81.s_jj $6$ (not in LMFDB) 2.81.bk_ss $6$ (not in LMFDB) 2.81.as_gg $12$ (not in LMFDB) 2.81.a_add $12$ (not in LMFDB) 2.81.a_gg $12$ (not in LMFDB) 2.81.s_gg $12$ (not in LMFDB) 2.81.a_a $24$ (not in LMFDB) 2.81.aj_dd $30$ (not in LMFDB) 2.81.j_dd $30$ (not in LMFDB)