Properties

Label 1.13.ac
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:  $1 - 2 x + 13 x^{2}$
Frobenius angles:  $\pm0.410543812489$
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
$A(\F_{q^r})$ 12 192 2268 28416 370092 4826304 62763132 815766528 10604381004 137857789632

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 12 192 2268 28416 370092 4826304 62763132 815766528 10604381004 137857789632

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.c$2$1.169.w
1.13.af$3$(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.c$2$1.169.w
1.13.af$3$(not in LMFDB)
1.13.h$3$(not in LMFDB)
1.13.ah$6$(not in LMFDB)
1.13.f$6$(not in LMFDB)