Properties

Label 2.256.ach_caz
Base Field $\F_{2^{8}}$
Dimension $2$
Ordinary Yes
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{2^{8}}$
Dimension:  $2$
L-polynomial:  $1 - 59 x + 1377 x^{2} - 15104 x^{3} + 65536 x^{4}$
Frobenius angles:  $\pm0.0363747766709$, $\pm0.176437257411$
Angle rank:  $2$ (numerical)
Number field:  4.0.1665657.4
Galois group:  $D_{4}$

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 a Jacobian, and hence is principally polarizable.

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 51751 4247566827 281358201468400 18446577982334005875 1208926029731150837928031 79228164488965816687424107200 5192296860188277101655497675123791 340282366867703868977100332234272387875 22300745198018608063440365321057209694047600 1461501637327836655848217480211972777528731851387

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 198 64810 16770255 4294928626 1099511818878 281474983726207 72057594060874158 18446744070823698466 4722366482761221803535 1208925819612092821932730

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{8}}$
The endomorphism algebra of this simple isogeny class is 4.0.1665657.4.
All geometric endomorphisms are defined over $\F_{2^{8}}$.

Base change

This is a primitive isogeny class.

Twists

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