Properties

Label 1.61.ao
Base Field $\F_{61}$
Dimension $1$
Ordinary Yes
$p$-rank $1$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{61}$
Dimension:  $1$
L-polynomial:  $1 - 14 x + 61 x^{2}$
Frobenius angles:  $\pm0.146275019398$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-3}) \)
Galois group:  $C_2$
Jacobians:  4

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 the Jacobians of 4 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 48 3648 226800 13847808 844634928 51520795200 3142746371568 191707336823808 11694146210737200 713342911860107328

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 48 3648 226800 13847808 844634928 51520795200 3142746371568 191707336823808 11694146210737200 713342911860107328

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{61}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}) \).
All geometric endomorphisms are defined over $\F_{61}$.

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon base change
1.61.o$2$(not in LMFDB)
1.61.b$3$(not in LMFDB)
1.61.n$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
1.61.o$2$(not in LMFDB)
1.61.b$3$(not in LMFDB)
1.61.n$3$(not in LMFDB)
1.61.an$6$(not in LMFDB)
1.61.ab$6$(not in LMFDB)