Properties

Label 1.13.af
Base Field $\F_{13}$
Dimension $1$
Ordinary Yes
$p$-rank $1$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{13}$
Dimension:  $1$
L-polynomial:  $1 - 5 x + 13 x^{2}$
Frobenius angles:  $\pm0.256122854178$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-3}) \)
Galois group:  $C_2$
Jacobians:  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 the Jacobians of 2 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 9 171 2268 28899 372069 4826304 62735913 815674275 10604381004 137858633811

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 9 171 2268 28899 372069 4826304 62735913 815674275 10604381004 137858633811

Decomposition and endomorphism algebra

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

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon base change
1.13.f$2$1.169.b
1.13.ac$3$(not in LMFDB)
1.13.h$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
1.13.f$2$1.169.b
1.13.ac$3$(not in LMFDB)
1.13.h$3$(not in LMFDB)
1.13.ah$6$(not in LMFDB)
1.13.c$6$(not in LMFDB)