# Properties

 Label 2.3.ab_g Base Field $\F_{3}$ Dimension $2$ Ordinary No $p$-rank $1$ Principally polarizable Yes Contains a Jacobian No

## Invariants

 Base field: $\F_{3}$ Dimension: $2$ L-polynomial: $( 1 - x + 3 x^{2} )( 1 + 3 x^{2} )$ 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 6 7 8 9 10 $A(\F_{q^r})$ 12 240 1008 4800 51972 564480 4968948 42720000 384782832 3487321200

 $r$ 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.
 Twist Extension Degree Common 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.
 Twist Extension Degree Common 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