# Properties

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

## Invariants

 Base field: $\F_{3}$ Dimension: $2$ L-polynomial: $( 1 - 2 x + 3 x^{2} )( 1 - x + 3 x^{2} )$ $1 - 3x + 8x^{2} - 9x^{3} + 9x^{4}$ Frobenius angles: $\pm0.304086723985$, $\pm0.406785250661$ Angle rank: $2$ (numerical) Jacobians: 0

This isogeny class is not simple.

## Newton polygon

This isogeny class is ordinary.

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

## Point counts

This isogeny class is principally polarizable, but does not contain a Jacobian.

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $6$ $180$ $1368$ $7200$ $51546$

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $1$ $17$ $46$ $89$ $211$ $674$ $2185$ $6641$ $19738$ $59057$

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ac $\times$ 1.3.ab and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
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_e$2$2.9.h_bc
2.3.b_e$2$2.9.h_bc
2.3.d_i$2$2.9.h_bc