# Properties

 Label 1.3.ad Base field $\F_{3}$ Dimension $1$ $p$-rank $0$ Ordinary no Supersingular yes Simple yes Geometrically simple yes Primitive yes Principally polarizable yes Contains a Jacobian yes

## Invariants

 Base field: $\F_{3}$ Dimension: $1$ L-polynomial: $1 - 3 x + 3 x^{2}$ Frobenius angles: $\pm0.166666666667$ 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 Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $1$ $7$ $28$ $91$ $271$

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $1$ $7$ $28$ $91$ $271$ $784$ $2269$ $6643$ $19684$ $58807$

## 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 is a list of all twists of this isogeny class.
TwistExtension degreeCommon base change
1.3.d$2$1.9.ad
1.3.a$3$1.27.a
1.3.d$3$1.27.a