# Properties

 Label 2.3.b_e Base Field $\F_{3}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable Yes Contains a Jacobian Yes

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## Invariants

 Base field: $\F_{3}$ Dimension: $2$ L-polynomial: $( 1 - x + 3 x^{2} )( 1 + 2 x + 3 x^{2} )$ Frobenius angles: $\pm0.406785250661$, $\pm0.695913276015$ Angle rank: $2$ (numerical) Jacobians: 1

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 contains the Jacobians of 1 curves, and hence is principally polarizable:

• $y^2=x^6+x^4+2x^3+x^2+2x$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 18 180 648 7200 52398 492480 5164254 43574400 381068712 3487086900

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 5 17 26 89 215 674 2357 6641 19358 59057

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ab $\times$ 1.3.c 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.
 Twist Extension Degree Common base change 2.3.ad_i $2$ 2.9.h_bc 2.3.ab_e $2$ 2.9.h_bc 2.3.d_i $2$ 2.9.h_bc