# Properties

 Label 2.3.b_d Base field $\F_{3}$ Dimension $2$ $p$-rank $1$ Ordinary no Supersingular no Simple yes Geometrically simple yes Primitive yes Principally polarizable yes Contains a Jacobian yes

## Invariants

 Base field: $\F_{3}$ Dimension: $2$ L-polynomial: $1 + x + 3 x^{2} + 3 x^{3} + 9 x^{4}$ Frobenius angles: $\pm0.377272149103$, $\pm0.731463671465$ Angle rank: $2$ (numerical) Number field: 4.0.11661.1 Galois group: $D_{4}$ Jacobians: 2

This isogeny class is simple and geometrically simple.

## Newton polygon

 $p$-rank: $1$ Slopes: $[0, 1/2, 1/2, 1]$

## Point counts

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

• $y^2=x^6+2x^4+2x^3+2$
• $y^2=2x^6+2x^4+2x^3+1$

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $17$ $153$ $731$ $8109$ $49232$

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $5$ $15$ $29$ $99$ $200$ $675$ $2315$ $6579$ $19847$ $58950$

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The endomorphism algebra of this simple isogeny class is 4.0.11661.1.
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.ab_d$2$2.9.f_v