Properties

Label 2.256.acg_bzv
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:  $1 - 58 x + 1347 x^{2} - 14848 x^{3} + 65536 x^{4}$
Frobenius angles:  $\pm0.0591285773829$, $\pm0.188509518148$
Angle rank:  $2$ (numerical)
Number field:  4.0.6419520.1
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
$A(\F_{q^r})$ 51978 4251280620 281386448471358 18446732588498124000 1208926707252173539280058 79228167008510659839499805580 5192296868971096393536151720157838 340282366903508113737635702969604144000 22300745198220240185173568724964982378966058 1461501637329167792156279981520434356091653435500

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 199 64867 16771939 4294964623 1099512435079 281474992677427 72057594182760259 18446744072764650463 4722366482803919066119 1208925819613193912079427

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{8}}$
The endomorphism algebra of this simple isogeny class is 4.0.6419520.1.
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_bzv$2$(not in LMFDB)