# Properties

 Label 2.3.ac_e Base field $\F_{3}$ Dimension $2$ $p$-rank $2$ Ordinary yes Supersingular no Simple yes Geometrically simple yes Primitive yes Principally polarizable yes Contains a Jacobian yes

# Related objects

## Invariants

 Base field: $\F_{3}$ Dimension: $2$ L-polynomial: $1 - 2 x + 4 x^{2} - 6 x^{3} + 9 x^{4}$ Frobenius angles: $\pm0.210767374595$, $\pm0.567777800232$ Angle rank: $2$ (numerical) Number field: 4.0.7488.1 Galois group: $D_{4}$ Jacobians: 2

This isogeny class is simple and geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

## Newton polygon

This isogeny class is ordinary. $p$-rank: $2$ Slopes: $[0, 0, 1, 1]$

## Point counts

This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=2x^6+x^5+2x^4+2x+1$
• $y^2=2x^5+2x^2+x$

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $6$ $132$ $702$ $6864$ $74526$

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $2$ $14$ $26$ $86$ $302$ $782$ $2102$ $6494$ $19682$ $58334$

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The endomorphism algebra of this simple isogeny class is 4.0.7488.1.
All geometric endomorphisms are defined over $\F_{3}$.

## Base change

This is a primitive isogeny class.

## Twists

Below is a list of all twists of this isogeny class.
TwistExtension degreeCommon base change
2.3.c_e$2$2.9.e_k