Properties

Label 2.1024.aes_inr
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:  $1 - 122 x + 5763 x^{2} - 124928 x^{3} + 1048576 x^{4}$
Frobenius angles:  $\pm0.0417801163030$, $\pm0.132307341722$
Angle rank:  $2$ (numerical)
Number field:  4.0.26989632.1
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
$A(\F_{q^r})$ 929290 1095995333100 1152834133349126590 1208924035784207482879200 1267650577085715370318651819450 1329227996012463897107521624543337100 1393796574937892098672597884867850381350670 1461501637332194008962688302242105962794437884800 1532495540865927711090121530762361037613044905019346730 1606938044258991052749328929402834054669544023038911096027500

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 903 1045219 1073660451 1099510005391 1125899886287943 1152921504804213427 1180591620742592027907 1208925819615697139951071 1237940039285411659885909959 1267650600228230014605210152899

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{10}}$
The endomorphism algebra of this simple isogeny class is 4.0.26989632.1.
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.es_inr$2$(not in LMFDB)