Properties

Label 2.1024.aer_ilh
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:  $1 - 121 x + 5701 x^{2} - 123904 x^{3} + 1048576 x^{4}$
Frobenius angles:  $\pm0.0506153084167$, $\pm0.141181143239$
Angle rank:  $2$ (numerical)
Number field:  4.0.65148065.1
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
$A(\F_{q^r})$ 930253 1096119900659 1152842254777899244 1208924430327096945344435 1267650592857840138572971008073 1329227996548605236795547661401657536 1393796574953387883048096491535529709575573 1461501637332562239805690983391281221699683675715 1532495540865934208740175985461892195396913587817328764 1606938044258991111484005614006063577206358824640340481811579

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 904 1045338 1073668015 1099510364226 1125899900296404 1152921505269241911 1180591620755717467816 1208925819616001733365346 1237940039285416908645824215 1267650600228230060938700370178

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{10}}$
The endomorphism algebra of this simple isogeny class is 4.0.65148065.1.
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_ilh$2$(not in LMFDB)