# Properties

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

## Invariants

 Base field: $\F_{3}$ Dimension: $2$ L-polynomial: $1 - 5 x^{2} + 9 x^{4}$ Frobenius angles: $\pm0.0932147493387$, $\pm0.906785250661$ Angle rank: $1$ (numerical) Number field: $$\Q(i, \sqrt{11})$$ Galois group: $C_2^2$ Jacobians: 0

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 is principally polarizable, but does not contain a Jacobian.

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 5 25 740 5625 59525 547600 4785485 44555625 387399620 3543225625

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 4 0 28 68 244 750 2188 6788 19684 60000

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The endomorphism algebra of this simple isogeny class is $$\Q(i, \sqrt{11})$$.
Endomorphism algebra over $\overline{\F}_{3}$
 The base change of $A$ to $\F_{3^{2}}$ is 1.9.af 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-11})$$$)$
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.ac_h $4$ 2.81.ao_id 2.3.a_f $4$ 2.81.ao_id 2.3.c_h $4$ 2.81.ao_id
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.3.ac_h $4$ 2.81.ao_id 2.3.a_f $4$ 2.81.ao_id 2.3.c_h $4$ 2.81.ao_id 2.3.ab_ac $12$ (not in LMFDB) 2.3.b_ac $12$ (not in LMFDB)