Properties

Label 2.1024.aew_ixl
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:  $( 1 - 63 x + 1024 x^{2} )^{2}$
Frobenius angles:  $\pm0.0563432964760$, $\pm0.0563432964760$
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
$A(\F_{q^r})$ 925444 1095488782336 1152800153499278596 1208922316379606963257344 1267650504532724409802515659524 1329227993372982245415435729892950016 1393796574855317554388413301161771679996164 1461501637330083485781733027850470549698086436864 1532495540865892738515237323732705906962768724238235396 1606938044258991405717506792985212383080903835299965529112576

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 899 1044735 1073628803 1099508441599 1125899821847939 1152921502514828031 1180591620672648668291 1208925819613951356093439 1237940039285383409264898947 1267650600228230293048006456575

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{10}}$
The isogeny class factors as 1.1024.acl 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-127}) \)$)$
All geometric endomorphisms are defined over $\F_{2^{10}}$.

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon base change
2.1024.a_acvx$2$(not in LMFDB)
2.1024.ew_ixl$2$(not in LMFDB)
2.1024.cl_ejh$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.1024.a_acvx$2$(not in LMFDB)
2.1024.ew_ixl$2$(not in LMFDB)
2.1024.cl_ejh$3$(not in LMFDB)
2.1024.a_cvx$4$(not in LMFDB)
2.1024.acl_ejh$6$(not in LMFDB)