Properties

Label 1.1024.acl
Base Field $\F_{2^{10}}$
Dimension $1$
Ordinary Yes
$p$-rank $1$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{2^{10}}$
Dimension:  $1$
L-polynomial:  $1 - 63 x + 1024 x^{2}$
Frobenius angles:  $\pm0.0563432964760$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-127}) \)
Galois group:  $C_2$

This isogeny class is simple and geometrically simple.

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $1$
Slopes:  $[0, 1]$

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
$A(\F_{q^r})$ 962 1046656 1073685314 1099510034688 1125899864345282 1152921503560837504 1180591620695029985858 1208925819614290265399808 1237940039285381842082011586 1267650600228229847272354830976

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 962 1046656 1073685314 1099510034688 1125899864345282 1152921503560837504 1180591620695029985858 1208925819614290265399808 1237940039285381842082011586 1267650600228229847272354830976

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{10}}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-127}) \).
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
1.1024.cl$2$(not in LMFDB)