# Properties

 Label 1.3.d Base Field $\F_{3}$ Dimension $1$ Ordinary No $p$-rank $0$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{3}$ Dimension: $1$ L-polynomial: $1 + 3 x + 3 x^{2}$ Frobenius angles: $\pm0.833333333333$ Angle rank: $0$ (numerical) Number field: $$\Q(\sqrt{-3})$$ Galois group: $C_2$ 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 $A(\F_{q^r})$ 7 7 28 91 217 784 2107 6643 19684 58807

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 7 7 28 91 217 784 2107 6643 19684 58807

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-3})$$.
Endomorphism algebra over $\overline{\F}_{3}$
 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 $\F_{3^{6}}$.
Remainder of endomorphism lattice by field
• Endomorphism algebra over $\F_{3^{2}}$  The base change of $A$ to $\F_{3^{2}}$ is the simple isogeny class 1.9.ad and its endomorphism algebra is $$\Q(\sqrt{-3})$$.
• Endomorphism algebra over $\F_{3^{3}}$  The base change of $A$ to $\F_{3^{3}}$ is the simple isogeny class 1.27.a and its endomorphism algebra is $$\Q(\sqrt{-3})$$.

## 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 1.3.ad $2$ 1.9.ad 1.3.ad $3$ 1.27.a 1.3.a $3$ 1.27.a
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 1.3.ad $2$ 1.9.ad 1.3.ad $3$ 1.27.a 1.3.a $3$ 1.27.a 1.3.a $6$ 1.729.cc