Properties

Label 2.4.ab_ac
Base Field $\F_{2^{2}}$
Dimension $2$
Ordinary No
$p$-rank $1$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{2^{2}}$
Dimension:  $2$
L-polynomial:  $1 - x - 2 x^{2} - 4 x^{3} + 16 x^{4}$
Frobenius angles:  $\pm0.123737125646$, $\pm0.736024913457$
Angle rank:  $2$ (numerical)
Number field:  4.0.8405.1
Galois group:  $D_{4}$
Jacobians:  2

This isogeny class is simple and geometrically simple.

Newton polygon

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

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
$A(\F_{q^r})$ 10 200 3070 74000 1038550 17007800 276074830 4296884000 69074008390 1102005405000

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 4 12 46 288 1014 4152 16846 65568 263494 1050952

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{2}}$
The endomorphism algebra of this simple isogeny class is 4.0.8405.1.
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_ac$2$2.16.af_bc