Properties

Label 2.1024.aer_ild
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:  $1 - 121 x + 5697 x^{2} - 123904 x^{3} + 1048576 x^{4}$
Frobenius angles:  $\pm0.0214972246866$, $\pm0.148663638065$
Angle rank:  $2$ (numerical)
Number field:  4.0.387225.1
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
$A(\F_{q^r})$ 930249 1096111466451 1152840695507517636 1208924273017241894269539 1267650581231639034226775345049 1329227995844220970512637693628695296 1393796574916401721762022784479130680512569 1461501637330825121405395250115378643642070773059 1532495540865859800297104658342794837453686803119433156 1606938044258988167456579671452432289375425717402734860449651

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 904 1045330 1073666563 1099510221154 1125899889970264 1152921504658285951 1180591620724388970616 1208925819614564822702914 1237940039285356801984184707 1267650600228227738510560145650

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{10}}$
The endomorphism algebra of this simple isogeny class is 4.0.387225.1.
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_ild$2$(not in LMFDB)