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:

Point counts of the abelian variety

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

Point counts of the curve

$r$ 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.
TwistExtension DegreeCommon 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