Properties

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

Learn more about

Invariants

Base field:  $\F_{2^{10}}$
Dimension:  $2$
L-polynomial:  $1 - 121 x + 5705 x^{2} - 123904 x^{3} + 1048576 x^{4}$
Frobenius angles:  $\pm0.0734698624279$, $\pm0.130488439581$
Angle rank:  $2$ (numerical)
Number field:  4.0.2618993.1
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
$A(\F_{q^r})$ 930257 1096128334899 1152843814048828148 1208924587566587439268227 1267650604462243841032602086297 1329227997249390063766652713552821504 1393796574989951564768308392279046174875761 1461501637334259815975388917555633522130541821123 1532495540866005481064365841508478940166285214630716116 1606938044258993836851167708914555814099936558265352980411459

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 904 1045346 1073669467 1099510507234 1125899910603184 1152921505877075855 1180591620786688110904 1208925819617405935461250 1237940039285474481970806091 1267650600228232210874306919986

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{10}}$
The endomorphism algebra of this simple isogeny class is 4.0.2618993.1.
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.er_ill$2$(not in LMFDB)