Properties

Label 2.4.ae_l
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:  $( 1 - 3 x + 4 x^{2} )( 1 - x + 4 x^{2} )$
Frobenius angles:  $\pm0.230053456163$, $\pm0.419569376745$
Angle rank:  $2$ (numerical)
Jacobians:  2

This isogeny class is not 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
$A(\F_{q^r})$ 8 384 5624 69120 1043048 17006976 271133528 4280186880 68254163144 1096502123904

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 1 23 85 271 1021 4151 16549 65311 260365 1045703

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{2}}$
The isogeny class factors as 1.4.ad $\times$ 1.4.ab and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
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.ac_f$2$2.16.g_z
2.4.c_f$2$2.16.g_z
2.4.e_l$2$2.16.g_z