Properties

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

Learn more about

Invariants

Base field:  $\F_{2^{8}}$
Dimension:  $2$
L-polynomial:  $1 - 57 x + 1317 x^{2} - 14592 x^{3} + 65536 x^{4}$
Frobenius angles:  $\pm0.0716566769452$, $\pm0.201368865130$
Angle rank:  $2$ (numerical)
Number field:  4.0.15360865.2
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
$A(\F_{q^r})$ 52205 4254864115 281411877547820 18446855403405581875 1208927131907351124367625 79228167947404481179410208960 5192296869570599837079810195062645 340282366902963055886432777798539831875 22300745198276544950996059160172166648920380 1461501637329919292145447938803799830291365362875

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 200 64922 16773455 4294993218 1099512821300 281474996013047 72057594191080040 18446744072735102818 4722366482815842064055 1208925819613815538300802

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{8}}$
The endomorphism algebra of this simple isogeny class is 4.0.15360865.2.
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.cf_byr$2$(not in LMFDB)