Properties

Label 2.1024.aes_inp
Base Field $\F_{2^{10}}$
Dimension $2$
Ordinary Yes
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{2^{10}}$
Dimension:  $2$
L-polynomial:  $1 - 122 x + 5761 x^{2} - 124928 x^{3} + 1048576 x^{4}$
Frobenius angles:  $\pm0.0233129119208$, $\pm0.136899915147$
Angle rank:  $2$ (numerical)
Number field:  4.0.999488.2
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
$A(\F_{q^r})$ 929288 1095991115744 1152833347268645576 1208923955545968593714048 1267650571063861667459848790088 1329227995640534199629433188873406176 1393796574917899240104359000073201516914312 1461501637331228322645181652356353024541669764608 1532495540865884957879518782469580380220606343708626312 1606938044258989294901829531090527858867500826257910884971744

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 903 1045215 1073659719 1099509932415 1125899880939463 1152921504481615839 1180591620725657417991 1208925819614898342951423 1237940039285377124117418759 1267650600228228627908040537055

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{10}}$
The endomorphism algebra of this simple isogeny class is 4.0.999488.2.
All geometric endomorphisms are defined over $\F_{2^{10}}$.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.1024.es_inp$2$(not in LMFDB)