Properties

Label 2.256.acg_bzt
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:  $1 - 58 x + 1345 x^{2} - 14848 x^{3} + 65536 x^{4}$
Frobenius angles:  $\pm0.0329768133648$, $\pm0.195159409513$
Angle rank:  $2$ (numerical)
Number field:  4.0.237632.1
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
$A(\F_{q^r})$ 51976 4251013088 281380604689864 18446663247467944832 1208926115449815756054856 79228162951175765410576885472 5192296845379730883249147125384584 340282366783544516310359738736423546368 22300745197675733500772164343784568565580424 1461501637326920211787655538758983881106089370848

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 199 64863 16771591 4294948479 1099511896839 281474978262879 72057593855364295 18446744066261410815 4722366482688615290695 1208925819611334757173343

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{8}}$
The endomorphism algebra of this simple isogeny class is 4.0.237632.1.
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_bzt$2$(not in LMFDB)