# Properties

 Label 1.27.a Base Field $\F_{3^{3}}$ Dimension $1$ Ordinary No $p$-rank $0$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{3^{3}}$ Dimension: $1$ Weil polynomial: $1 + 27 x^{2}$ Frobenius angles: $\pm0.5$ Angle rank: $0$ (numerical) Number field: $$\Q(\sqrt{-3})$$ Galois group: $C_2$ Jacobians: 2

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 2 curves, and hence is principally polarizable:

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 28 784 19684 529984 14348908 387459856 10460353204 282428473600 7625597484988 205891160792464

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 28 784 19684 529984 14348908 387459856 10460353204 282428473600 7625597484988 205891160792464

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{3}}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-3})$$.
Endomorphism algebra over $\overline{\F}_{3^{3}}$
 The base change of $A$ to $\F_{3^{6}}$ is the simple isogeny class 1.729.cc and its endomorphism algebra is the quaternion algebra over $$\Q$$ ramified at $3$ and $\infty$.
All geometric endomorphisms are defined over $\F_{3^{6}}$.

## Base change

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

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

## Twists

Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 1.27.aj $3$ (not in LMFDB) 1.27.j $3$ (not in LMFDB) 1.27.j $6$ (not in LMFDB)