Properties

Label 1.317.aj
Base Field $\F_{317}$
Dimension $1$
Ordinary Yes
$p$-rank $1$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{317}$
Dimension:  $1$
L-polynomial:  $1 - 9 x + 317 x^{2}$
Frobenius angles:  $\pm0.418666569573$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-1187}) \)
Galois group:  $C_2$
Jacobians:  9

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 9 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 309 101043 31862844 10097934291 3201074975769 1014741855631296 321673168581485421 101970394098452992323 32324614926022899918828 10246902931628954276247243

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 309 101043 31862844 10097934291 3201074975769 1014741855631296 321673168581485421 101970394098452992323 32324614926022899918828 10246902931628954276247243

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{317}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-1187}) \).
All geometric endomorphisms are defined over $\F_{317}$.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
1.317.j$2$(not in LMFDB)