Properties

Label 2.1024.d_absf
Base field $\F_{2^{10}}$
Dimension $2$
$p$-rank $2$
Ordinary Yes
Supersingular No
Simple Yes
Geometrically simple Yes
Primitive Yes
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{2^{10}}$
Dimension:  $2$
L-polynomial:  $1 + 3 x - 1149 x^{2} + 3072 x^{3} + 1048576 x^{4}$
Frobenius angles:  $\pm0.170249732133$, $\pm0.861797799942$
Angle rank:  $2$ (numerical)
Number field:  4.0.126316622340433.1
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
$A(\F_{q^r})$ 1050503 1097095959559 1152942534952668956 1208927442048880663137883 1267650625016731908452493777113 1329228000383123440463594155327000336 1393796574894308318075178919059535925417147 1461501637334533164636778167800126534414802827763 1532495540865817266243154932468519680584937766730429412 1606938044258989613574920183674983076866516080184677216782519

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 1028 1046270 1073761409 1099513103370 1125899928859238 1152921508595156279 1180591620705675129776 1208925819617632044174418 1237940039285322443249027969 1267650600228228879296783298830

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{10}}$
The endomorphism algebra of this simple isogeny class is 4.0.126316622340433.1.
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.ad_absf$2$(not in LMFDB)