# Properties

 Label 2.3.ab_g Base field $\F_{3}$ Dimension $2$ $p$-rank $1$ Ordinary no 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: $( 1 - x + 3 x^{2} )( 1 + 3 x^{2} )$ $1 - x + 6 x^{2} - 3 x^{3} + 9 x^{4}$ Frobenius angles: $\pm0.406785250661$, $\pm0.5$ Angle rank: $1$ (numerical) Jacobians: 0

This isogeny class is not simple.

## Newton polygon

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

## Point counts

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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $12$ $240$ $1008$ $4800$ $51972$

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $3$ $21$ $36$ $57$ $213$ $774$ $2271$ $6513$ $19548$ $59061$

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ab $\times$ 1.3.a 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 $\times$ 1.9.g. The endomorphism algebra for each factor is: 1.9.f : $$\Q(\sqrt{-11})$$. 1.9.g : the quaternion algebra over $$\Q$$ ramified at $3$ and $\infty$.
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.b_g$2$2.9.l_bw
2.3.ae_j$3$2.27.i_cc
2.3.c_d$3$2.27.i_cc
Below is a list of all twists of this isogeny class.
TwistExtension degreeCommon base change
2.3.b_g$2$2.9.l_bw
2.3.ae_j$3$2.27.i_cc
2.3.c_d$3$2.27.i_cc
2.3.ae_j$6$2.729.bs_bji
2.3.ac_d$6$2.729.bs_bji
2.3.c_d$6$2.729.bs_bji
2.3.e_j$6$2.729.bs_bji