Properties

Label 1.169.aw
Base Field $\F_{13^{2}}$
Dimension $1$
Ordinary Yes
$p$-rank $1$
Principally polarizable Yes
Contains a Jacobian Yes

Learn more about

Invariants

Base field:  $\F_{13^{2}}$
Dimension:  $1$
L-polynomial:  $1 - 22 x + 169 x^{2}$
Frobenius angles:  $\pm0.178912375022$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-3}) \)
Galois group:  $C_2$
Jacobians:  8

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 8 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 148 28416 4827316 815766528 137859194068 23298094520064 3937376473771252 665416609532571648 112455406944759851284 19004963774663406302976

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 148 28416 4827316 815766528 137859194068 23298094520064 3937376473771252 665416609532571648 112455406944759851284 19004963774663406302976

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13^{2}}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}) \).
All geometric endomorphisms are defined over $\F_{13^{2}}$.

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon base change
1.169.w$2$(not in LMFDB)
1.169.ab$3$(not in LMFDB)
1.169.x$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
1.169.w$2$(not in LMFDB)
1.169.ab$3$(not in LMFDB)
1.169.x$3$(not in LMFDB)
1.169.ax$6$(not in LMFDB)
1.169.b$6$(not in LMFDB)