# Properties

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

## Invariants

 Base field: $\F_{3}$ Dimension: $2$ L-polynomial: $( 1 - 2 x + 3 x^{2} )( 1 + 3 x + 3 x^{2} )$ Frobenius angles: $\pm0.304086723985$, $\pm0.833333333333$ Angle rank: $1$ (numerical) Jacobians: 1

This isogeny class is not simple.

## Newton polygon

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

## Point counts

This isogeny class contains the Jacobians of 1 curves, and hence is principally polarizable:

• $y^2=x^6+2x^4+x$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 14 84 1064 8736 52514 536256 4428914 43365504 391199816 3500898324

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 5 9 38 105 215 738 2021 6609 19874 59289

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ac $\times$ 1.3.d 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.
 Twist Extension Degree Common base change 2.3.af_m $2$ 2.9.ab_m 2.3.ab_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.
 Twist Extension Degree Common base change 2.3.af_m $2$ 2.9.ab_m 2.3.ab_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.c_g $6$ 2.729.i_abnm