# Properties

 Label 2.3.a_c Base field $\F_{3}$ Dimension $2$ $p$-rank $2$ Ordinary yes Supersingular no Simple no Geometrically simple no Primitive yes Principally polarizable yes Contains a Jacobian yes

## Invariants

 Base field: $\F_{3}$ Dimension: $2$ L-polynomial: $( 1 - 2 x + 3 x^{2} )( 1 + 2 x + 3 x^{2} )$ $1 + 2 x^{2} + 9 x^{4}$ Frobenius angles: $\pm0.304086723985$, $\pm0.695913276015$ Angle rank: $1$ (numerical) Jacobians: 2

This isogeny class is not 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:

• $y^2=x^5+x$
• $y^2=x^6+x^5+2x^3+x+2$

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $12$ $144$ $684$ $9216$ $59532$

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $4$ $14$ $28$ $110$ $244$ $638$ $2188$ $6494$ $19684$ $60014$

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ac $\times$ 1.3.c and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
Endomorphism algebra over $\overline{\F}_{3}$
 The base change of $A$ to $\F_{3^{2}}$ is 1.9.c 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$2$2.9.e_w
2.3.e_k$2$2.9.e_w
2.3.a_ac$4$2.81.bc_nu
Below is a list of all twists of this isogeny class.
TwistExtension degreeCommon base change
2.3.ae_k$2$2.9.e_w
2.3.e_k$2$2.9.e_w
2.3.a_ac$4$2.81.bc_nu
2.3.ac_b$6$2.729.ado_fhm
2.3.c_b$6$2.729.ado_fhm
2.3.ae_i$8$(not in LMFDB)
2.3.e_i$8$(not in LMFDB)