# Properties

 Label 2.3.b_ac Base Field $\F_{3}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{3}$ Dimension: $2$ L-polynomial: $1 + x - 2 x^{2} + 3 x^{3} + 9 x^{4}$ Frobenius angles: $\pm0.259881416005$, $\pm0.926548082672$ Angle rank: $1$ (numerical) Number field: $$\Q(\sqrt{-3}, \sqrt{-11})$$ Galois group: $C_2^2$ Jacobians: 1

This isogeny class is simple but not geometrically 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 1 curves, and hence is principally polarizable:

• $y^2=x^5+2x^4+2x^3+2x$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 12 48 1296 7104 67332 518400 4606068 42311424 382124304 3514999728

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 5 5 44 89 275 710 2105 6449 19412 59525

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-3}, \sqrt{-11})$$.
Endomorphism algebra over $\overline{\F}_{3}$
 The base change of $A$ to $\F_{3^{3}}$ is 1.27.i 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-11})$$$)$
All geometric endomorphisms are defined over $\F_{3^{3}}$.

## 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.ab_ac $2$ 2.9.af_q 2.3.ac_h $3$ 2.27.q_eo 2.3.a_f $6$ 2.729.au_chy
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.3.ab_ac $2$ 2.9.af_q 2.3.ac_h $3$ 2.27.q_eo 2.3.a_f $6$ 2.729.au_chy 2.3.c_h $6$ 2.729.au_chy 2.3.a_af $12$ (not in LMFDB)