Properties

Label 2.256.ack_cer
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:  $( 1 - 31 x + 256 x^{2} )^{2}$
Frobenius angles:  $\pm0.0797861753495$, $\pm0.0797861753495$
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
$A(\F_{q^r})$ 51076 4236447744 281274289882756 18446138247776845824 1208924379804581299596676 79228161252211611842173440000 5192296872675221101264372445983876 340282367054330365605229844832623591424 22300745199351984843516283006769043752360836 1461501637335183299916426696252531589500595113984

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 195 64639 16765251 4294826239 1099510318275 281474972226943 72057594234165315 18446744080940741119 4722366483043575319491 1208925819618169823473279

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{8}}$
The isogeny class factors as 1.256.abf 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-7}) \)$)$
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_arh$2$(not in LMFDB)
2.256.ck_cer$2$(not in LMFDB)
2.256.bf_bbd$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.256.a_arh$2$(not in LMFDB)
2.256.ck_cer$2$(not in LMFDB)
2.256.bf_bbd$3$(not in LMFDB)
2.256.a_rh$4$(not in LMFDB)
2.256.abf_bbd$6$(not in LMFDB)