Properties

Label 2.64.az_kp
Base Field $\F_{2^{6}}$
Dimension $2$
Ordinary Yes
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{2^{6}}$
Dimension:  $2$
L-polynomial:  $1 - 25 x + 275 x^{2} - 1600 x^{3} + 4096 x^{4}$
Frobenius angles:  $\pm0.0763965765331$, $\pm0.298668725704$
Angle rank:  $2$ (numerical)
Number field:  4.0.3297921.2
Galois group:  $D_{4}$
Jacobians:  12

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 12 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 2747 16473759 68771683412 281508684183531 1152897481748456477 4722333178141140465744 19342797421940923903701767 79228162528834195562225281875 324518559038992552084565483210732 1329228000084445085013278452474245279

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 40 4022 262345 16779226 1073719450 68718992087 4398042943180 281474976762418 18014398808163385 1152921508336094102

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{6}}$
The endomorphism algebra of this simple isogeny class is 4.0.3297921.2.
All geometric endomorphisms are defined over $\F_{2^{6}}$.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.64.z_kp$2$(not in LMFDB)