Properties

Label 2.256.acg_bzz
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:  $1 - 58 x + 1351 x^{2} - 14848 x^{3} + 65536 x^{4}$
Frobenius angles:  $\pm0.100628845752$, $\pm0.169176663725$
Angle rank:  $2$ (numerical)
Number field:  4.0.1666112.1
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
$A(\F_{q^r})$ 51982 4251815708 281398136131378 18446871064572620768 1208927883204478915070782 79228174973278286216762677148 5192296914088865540202522940929058 340282367121007860494572325030644065152 22300745199105339882282243520522262839882798 1461501637332054401357041105688503752611534070108

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 199 64875 16772635 4294996863 1099513504599 281475020973963 72057594808895083 18446744084555335359 4722366482991346217383 1208925819615581659246315

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{8}}$
The endomorphism algebra of this simple isogeny class is 4.0.1666112.1.
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.cg_bzz$2$(not in LMFDB)