Properties

Label 2.1024.aer_ilf
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:  $1 - 121 x + 5699 x^{2} - 123904 x^{3} + 1048576 x^{4}$
Frobenius angles:  $\pm0.0381294656189$, $\pm0.145171135660$
Angle rank:  $2$ (numerical)
Number field:  4.0.63621537.2
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
$A(\F_{q^r})$ 930251 1096115683551 1152841475142640052 1208924351680964989998123 1267650587047464261641567802941 1329227996196863088908341180280055504 1393796574934947636349859037189139045629623 1461501637331698628857958411808315572413544779347 1532495540865897397407899784147115266591793412263399628 1606938044258989666912684249947686763154183196642554191687231

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 904 1045334 1073667289 1099510292698 1125899895135754 1152921504964154231 1180591620740097971308 1208925819615287371132594 1237940039285387172688420777 1267650600228228921372860142614

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{10}}$
The endomorphism algebra of this simple isogeny class is 4.0.63621537.2.
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.er_ilf$2$(not in LMFDB)