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.

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 929296 1096007985216 1152836491591399696 1208924276393377476000000 1267650595118310161442858276496 1329227997122753830781873639035223616 1393796574997217392814291439349248162161296 1461501637335029060117379176280120534656016000000 1532495540866050973276803398598699779237508846553831696 1606938044258995971477084063420559560539245678925878391486016

Point counts of the curve

$r$ 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 $\F_{2^{10}}$.

SubfieldPrimitive Model
$\F_{2}$2.2.a_ab

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon 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.
TwistExtension DegreeCommon 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)