# Properties

 Label 2.3.a_e 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: $1 + 4 x^{2} + 9 x^{4}$ Frobenius angles: $\pm0.366139763599$, $\pm0.633860236401$ Angle rank: $1$ (numerical) Number field: $$\Q(\sqrt{2}, \sqrt{-5})$$ 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:

• $y^2=x^6+2x^5+x+2$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 14 196 686 7056 58814 470596 4787006 45158400 387431534 3459086596

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 4 18 28 86 244 642 2188 6878 19684 58578

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{2}, \sqrt{-5})$$.
Endomorphism algebra over $\overline{\F}_{3}$
 The base change of $A$ to $\F_{3^{2}}$ is 1.9.e 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-5})$$$)$
All geometric endomorphisms are defined over $\F_{3^{2}}$.

## Base change

This is a primitive isogeny class.

## Twists

Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.3.a_ae $4$ 2.81.e_gk 2.3.ac_c $8$ (not in LMFDB) 2.3.c_c $8$ (not in LMFDB)