# Properties

 Label 2.3.af_m 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

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## Invariants

 Base field: $\F_{3}$ Dimension: $2$ L-polynomial: $( 1 - 3 x + 3 x^{2} )( 1 - 2 x + 3 x^{2} )$ $1 - 5x + 12x^{2} - 15x^{3} + 9x^{4}$ Frobenius angles: $\pm0.166666666667$, $\pm0.304086723985$ 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.

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $2$ $84$ $1064$ $8736$ $65582$

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $-1$ $9$ $38$ $105$ $269$ $738$ $2183$ $6609$ $19874$ $59289$

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ad $\times$ 1.3.ac 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 $\F_{3^{6}}$ is 1.729.abu $\times$ 1.729.cc. The endomorphism algebra for each factor is: 1.729.abu : $$\Q(\sqrt{-2})$$. 1.729.cc : the quaternion algebra over $$\Q$$ ramified at $3$ and $\infty$.
All geometric endomorphisms are defined over $\F_{3^{6}}$.
Remainder of endomorphism lattice by field
• Endomorphism algebra over $\F_{3^{2}}$  The base change of $A$ to $\F_{3^{2}}$ is 1.9.ad $\times$ 1.9.c. The endomorphism algebra for each factor is:
• Endomorphism algebra over $\F_{3^{3}}$  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a $\times$ 1.27.k. The endomorphism algebra for each factor is:

## 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.ab_a$2$2.9.ab_m
2.3.b_a$2$2.9.ab_m
2.3.f_m$2$2.9.ab_m
2.3.ac_g$3$2.27.k_cc
Below is a list of all twists of this isogeny class.
TwistExtension degreeCommon base change
2.3.ab_a$2$2.9.ab_m
2.3.b_a$2$2.9.ab_m
2.3.f_m$2$2.9.ab_m
2.3.ac_g$3$2.27.k_cc
2.3.ab_a$6$2.729.i_abnm
2.3.c_g$6$2.729.i_abnm