Properties

Label 2.1024.aes_inv
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:  $1 - 122 x + 5767 x^{2} - 124928 x^{3} + 1048576 x^{4}$
Frobenius angles:  $\pm0.0710065369012$, $\pm0.118913228236$
Angle rank:  $2$ (numerical)
Number field:  4.0.6999104.2
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
$A(\F_{q^r})$ 929294 1096003767836 1152835705510503602 1208924196207911922915296 1267650589112939617298113203774 1329227996753573602003192665739317404 1393796574977551126632751947387515055095714 1461501637334094366555934026267221528371984171904 1532495540866010716989497497180402788148435637273817902 1606938044258994390808175834713927712146909998999279984170076

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 903 1045227 1073661915 1099510151295 1125899896970263 1152921505447023627 1180591620776184531243 1208925819617269078909119 1237940039285478711517444455 1267650600228232647869333803627

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{10}}$
The endomorphism algebra of this simple isogeny class is 4.0.6999104.2.
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_inv$2$(not in LMFDB)