Properties

Label 2.256.acg_bzx
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:  $( 1 - 31 x + 256 x^{2} )( 1 - 27 x + 256 x^{2} )$
Frobenius angles:  $\pm0.0797861753495$, $\pm0.180343027596$
Angle rank:  $2$ (numerical)

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 a Jacobian, and hence is principally polarizable.

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 51980 4251548160 281392292285180 18446801860866263040 1208927296503727709019500 79228171015869159229184064000 5192296891873673192653689090467420 340282367015983191966168674620574146560 22300745198696535948327624892870249642332620 1461501637330875658776560408980422547339665984000

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 199 64871 16772287 4294980751 1099512970999 281475006914423 72057594500597359 18446744078861936671 4722366482904778617127 1208925819614606626236551

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{8}}$
The isogeny class factors as 1.256.abf $\times$ 1.256.abb 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 $\F_{2^{8}}$.

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.ae_amn$2$(not in LMFDB)
2.256.e_amn$2$(not in LMFDB)
2.256.cg_bzx$2$(not in LMFDB)