# 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

## 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.

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 930249 1096111466451 1152840695507517636 1208924273017241894269539 1267650581231639034226775345049 1329227995844220970512637693628695296 1393796574916401721762022784479130680512569 1461501637330825121405395250115378643642070773059 1532495540865859800297104658342794837453686803119433156 1606938044258988167456579671452432289375425717402734860449651

 $r$ 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.
 Twist Extension Degree Common base change 2.1024.er_ild $2$ (not in LMFDB)