Properties

Label 1.313.an
Base Field $\F_{313}$
Dimension $1$
Ordinary Yes
$p$-rank $1$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{313}$
Dimension:  $1$
L-polynomial:  $1 - 13 x + 313 x^{2}$
Frobenius angles:  $\pm0.380247423634$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-3}) \)
Galois group:  $C_2$
Jacobians:  7

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 7 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 301 98427 30674308 9597912051 3004147211821 940299071632704 294313622115692389 92120163576016007523 28833611193419038688164 9024920303509556734939707

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 301 98427 30674308 9597912051 3004147211821 940299071632704 294313622115692389 92120163576016007523 28833611193419038688164 9024920303509556734939707

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{313}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}) \).
All geometric endomorphisms are defined over $\F_{313}$.

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon base change
1.313.n$2$(not in LMFDB)
1.313.aw$3$(not in LMFDB)
1.313.bj$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
1.313.n$2$(not in LMFDB)
1.313.aw$3$(not in LMFDB)
1.313.bj$3$(not in LMFDB)
1.313.abj$6$(not in LMFDB)
1.313.w$6$(not in LMFDB)