Properties

Label 2.256.acg_cab
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:  $( 1 - 29 x + 256 x^{2} )^{2}$
Frobenius angles:  $\pm0.138932406859$, $\pm0.138932406859$
Angle rank:  $1$ (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
$A(\F_{q^r})$ 51984 4252083264 281403980010000 18446940199617456384 1208928467354427809821584 79228178880765063447715560000 5192296935618077695800794316983184 340282367218620214654387395473948795904 22300745199447370132496450337670177013610000 1461501637332714576827161572334880512526343672384

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 199 64879 16772983 4295012959 1099514035879 281475034856143 72057595107672919 18446744089846911679 4722366483063773940103 1208925819616127743596079

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{8}}$
The isogeny class factors as 1.256.abd 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-183}) \)$)$
All geometric endomorphisms are defined over $\F_{2^{8}}$.

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon base change
2.256.a_amr$2$(not in LMFDB)
2.256.cg_cab$2$(not in LMFDB)
2.256.bd_wn$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.256.a_amr$2$(not in LMFDB)
2.256.cg_cab$2$(not in LMFDB)
2.256.bd_wn$3$(not in LMFDB)
2.256.a_mr$4$(not in LMFDB)
2.256.abd_wn$6$(not in LMFDB)