# Properties

 Label 2.2.b_a Base field $\F_{2}$ Dimension $2$ $p$-rank $1$ Ordinary no Supersingular no Simple yes Geometrically simple yes Primitive yes Principally polarizable yes Contains a Jacobian yes

# Related objects

## Invariants

 Base field: $\F_{2}$ Dimension: $2$ L-polynomial: $1 + x + 2 x^{3} + 4 x^{4}$ Frobenius angles: $\pm0.313829601922$, $\pm0.860613258134$ Angle rank: $2$ (numerical) Number field: 4.0.2312.1 Galois group: $D_{4}$ Jacobians: 1

This isogeny class is simple and geometrically simple, primitive, not ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

## Newton polygon $p$-rank: $1$ Slopes: $[0, 1/2, 1/2, 1]$

## Point counts

This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:

• $y^2+xy=x^5+x$

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $8$ $16$ $152$ $416$ $808$

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $4$ $4$ $16$ $24$ $24$ $64$ $88$ $288$ $520$ $1104$

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
 The endomorphism algebra of this simple isogeny class is 4.0.2312.1.
All geometric endomorphisms are defined over $\F_{2}$.

## Base change

This is a primitive isogeny class.

## Twists

Below is a list of all twists of this isogeny class.
TwistExtension degreeCommon base change
2.2.ab_a$2$2.4.ab_e