# Properties

 Label 2.3.ab_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: $\pm0.0734519173280$, $\pm0.740118583995$ 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:

• $y^2=2x^5+x^4+x^3+x$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 4 48 400 7104 52204 518400 4969276 42311424 392832400 3514999728

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 3 5 12 89 213 710 2271 6449 19956 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.ai 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.b_ac $2$ 2.9.af_q 2.3.c_h $3$ 2.27.aq_eo 2.3.ac_h $6$ 2.729.au_chy
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.3.b_ac $2$ 2.9.af_q 2.3.c_h $3$ 2.27.aq_eo 2.3.ac_h $6$ 2.729.au_chy 2.3.a_f $6$ 2.729.au_chy 2.3.a_af $12$ (not in LMFDB)