# Properties

 Label 2.3.a_e Base field $\F_{3}$ Dimension $2$ $p$-rank $2$ Ordinary yes Supersingular no Simple yes Geometrically simple no Primitive yes 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 Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:

• $y^2=x^6+2x^5+x+2$

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $14$ $196$ $686$ $7056$ $58814$

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

## 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.
TwistExtension degreeCommon 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)