Properties

Label 2.3.a_f
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:  $( 1 - x + 3 x^{2} )( 1 + x + 3 x^{2} )$
Frobenius angles:  $\pm0.406785250661$, $\pm0.593214749339$
Angle rank:  $1$ (numerical)
Jacobians:  2

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 2 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 15 225 720 5625 58575 518400 4780455 44555625 387441360 3431030625

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 4 20 28 68 244 710 2188 6788 19684 58100

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
The isogeny class factors as 1.3.ab $\times$ 1.3.b and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
Endomorphism algebra over $\overline{\F}_{3}$
The base change of $A$ to $\F_{3^{2}}$ is 1.9.f 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-11}) \)$)$
All geometric endomorphisms are defined over $\F_{3^{2}}$.

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon base change
2.3.ac_h$2$2.9.k_br
2.3.c_h$2$2.9.k_br
2.3.a_af$4$2.81.ao_id
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.3.ac_h$2$2.9.k_br
2.3.c_h$2$2.9.k_br
2.3.a_af$4$2.81.ao_id
2.3.ab_ac$6$2.729.au_chy
2.3.b_ac$6$2.729.au_chy