# Properties

 Label 2.3.a_ac 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: $1 - 2 x^{2} + 9 x^{4}$ Frobenius angles: $\pm0.195913276015$, $\pm0.804086723985$ Angle rank: $1$ (numerical) Number field: $$\Q(\zeta_{8})$$ Galois group: $C_2^2$ Jacobians: 2

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 2 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=x^5+2x$
• $y^2=x^5+x^4+2x^2+x+2$

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $8$ $64$ $776$ $9216$ $58568$

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $4$ $6$ $28$ $110$ $244$ $822$ $2188$ $6494$ $19684$ $58086$

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\zeta_{8})$$.
Endomorphism algebra over $\overline{\F}_{3}$
 The base change of $A$ to $\F_{3^{2}}$ is 1.9.ac 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-2})$$$)$
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.
TwistExtension degreeCommon base change
2.3.ae_k$4$2.81.bc_nu
2.3.a_c$4$2.81.bc_nu
2.3.e_k$4$2.81.bc_nu
Below is a list of all twists of this isogeny class.
TwistExtension degreeCommon base change
2.3.ae_k$4$2.81.bc_nu
2.3.a_c$4$2.81.bc_nu
2.3.e_k$4$2.81.bc_nu
2.3.ae_i$8$(not in LMFDB)
2.3.e_i$8$(not in LMFDB)
2.3.ac_b$12$(not in LMFDB)
2.3.c_b$12$(not in LMFDB)