Properties

Label 2.4.ab_f
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:  $1 - x + 5 x^{2} - 4 x^{3} + 16 x^{4}$
Frobenius angles:  $\pm0.304731991158$, $\pm0.605597892078$
Angle rank:  $2$ (numerical)
Number field:  4.0.2873.1
Galois group:  $D_{4}$
Jacobians:  3

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 3 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 17 459 4148 70227 1110797 16127424 261613901 4318188003 68947130996 1099519078059

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 4 26 67 274 1084 3935 15964 65890 263011 1048586

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{2}}$
The endomorphism algebra of this simple isogeny class is 4.0.2873.1.
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_f$2$2.16.j_bx