Properties

Label 1.1024.acm
Base Field $\F_{2^{10}}$
Dimension $1$
Ordinary No
$p$-rank $0$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{2^{10}}$
Dimension:  $1$
L-polynomial:  $( 1 - 32 x )^{2}$
Frobenius angles:  $0$, $0$
Angle rank:  $0$ (numerical)
Number field:  \(\Q\)
Galois group:  Trivial

This isogeny class is simple and geometrically simple.

Newton polygon

This isogeny class is supersingular.

$p$-rank:  $0$
Slopes:  $[1/2, 1/2]$

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
$A(\F_{q^r})$ 961 1046529 1073676289 1099509530625 1125899839733761 1152921502459363329 1180591620648691826689 1208925819612430151450625 1237940039285309906154946561 1267650600228227149696889520129

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 961 1046529 1073676289 1099509530625 1125899839733761 1152921502459363329 1180591620648691826689 1208925819612430151450625 1237940039285309906154946561 1267650600228227149696889520129

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{10}}$
The endomorphism algebra of this simple isogeny class is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
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
1.1024.cm$2$(not in LMFDB)
1.1024.bg$3$(not in LMFDB)
1.1024.a$4$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
1.1024.cm$2$(not in LMFDB)
1.1024.bg$3$(not in LMFDB)
1.1024.a$4$(not in LMFDB)
1.1024.abg$6$(not in LMFDB)