Properties

Label 2.1024.aeo_idz
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:  $1 - 118 x + 5511 x^{2} - 120832 x^{3} + 1048576 x^{4}$
Frobenius angles:  $\pm0.0490179712428$, $\pm0.173198423717$
Angle rank:  $2$ (numerical)
Number field:  4.0.6769728.2
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
$A(\F_{q^r})$ 933138 1096472609244 1152862843706930286 1208925250329445072805088 1267650614943247340626217122338 1329227996749942026292650540392994396 1393796574933509459769166868966661048538302 1461501637330954371125430872667172563240473040768 1532495540865856850861052548011877254568353283014937202 1606938044258988286468557625921048133070359336499442881737244

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 907 1045675 1073687191 1099511110015 1125899919912187 1152921505443873739 1180591620738879788359 1208925819614671735565503 1237940039285354419448715691 1267650600228227832394456806955

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{10}}$
The endomorphism algebra of this simple isogeny class is 4.0.6769728.2.
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.eo_idz$2$(not in LMFDB)