Properties

Label 2.4.ab_b
Base Field $\F_{2^{2}}$
Dimension $2$
Ordinary Yes
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{2^{2}}$
Dimension:  $2$
L-polynomial:  $1 - x + x^{2} - 4 x^{3} + 16 x^{4}$
Frobenius angles:  $\pm0.205814536801$, $\pm0.684666034597$
Angle rank:  $2$ (numerical)
Number field:  4.0.54665.1
Galois group:  $D_{4}$
Jacobians:  2

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 Jacobians of 2 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 13 299 3484 79235 1132573 16667456 272590513 4282572515 68246337484 1099301402979

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 4 18 55 306 1104 4071 16636 65346 260335 1048378

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{2}}$
The endomorphism algebra of this simple isogeny class is 4.0.54665.1.
All geometric endomorphisms are defined over $\F_{2^{2}}$.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.4.b_b$2$2.16.b_z