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

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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:

Point counts of the abelian variety

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

Point counts of the curve

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

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