# Properties

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

## Invariants

 Base field: $\F_{2}$ Dimension: $2$ L-polynomial: $1 - x + 3 x^{2} - 2 x^{3} + 4 x^{4}$ Frobenius angles: $\pm0.306143893905$, $\pm0.570118980449$ Angle rank: $2$ (numerical) Number field: 4.0.1025.1 Galois group: $D_{4}$ Jacobians: 1

This isogeny class is simple and 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:

• $y^2+(x^3+x+1)y=x^5+x^4+1$

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $5$ $55$ $80$ $275$ $1375$

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $2$ $10$ $11$ $18$ $42$ $55$ $86$ $258$ $587$ $1050$

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
 The endomorphism algebra of this simple isogeny class is 4.0.1025.1.
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.b_d$2$2.4.f_n