# Properties

 Label 2.1024.aes_inx 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: $( 1 - 61 x + 1024 x^{2} )^{2}$ Frobenius angles: $\pm0.0978468837242$, $\pm0.0978468837242$ Angle rank: $1$ (numerical)

This isogeny class is not 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 $A(\F_{q^r})$ 929296 1096007985216 1152836491591399696 1208924276393377476000000 1267650595118310161442858276496 1329227997122753830781873639035223616 1393796574997217392814291439349248162161296 1461501637335029060117379176280120534656016000000 1532495540866050973276803398598699779237508846553831696 1606938044258995971477084063420559560539245678925878391486016

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 903 1045231 1073662647 1099510224223 1125899902304103 1152921505767236431 1180591620792842506647 1208925819618042239301823 1237940039285511230287359303 1267650600228233894797239603631

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{10}}$
 The isogeny class factors as 1.1024.acj 2 and its endomorphism algebra is $\mathrm{M}_{2}($$$\Q(\sqrt{-15})$$$)$
All geometric endomorphisms are defined over $\F_{2^{10}}$.

## Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{2^{10}}$.

 Subfield Primitive Model $\F_{2}$ 2.2.a_ab

## Twists

Below are some of the twists of this isogeny class.
 Twist Extension Degree Common base change 2.1024.a_acmj $2$ (not in LMFDB) 2.1024.es_inx $2$ (not in LMFDB) 2.1024.cj_dzt $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.1024.a_acmj $2$ (not in LMFDB) 2.1024.es_inx $2$ (not in LMFDB) 2.1024.cj_dzt $3$ (not in LMFDB) 2.1024.a_cmj $4$ (not in LMFDB) 2.1024.acj_dzt $6$ (not in LMFDB)