Properties

Label 1.337.ap
Base Field $\F_{337}$
Dimension $1$
Ordinary Yes
$p$-rank $1$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{337}$
Dimension:  $1$
L-polynomial:  $1 - 15 x + 337 x^{2}$
Frobenius angles:  $\pm0.366034192158$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-1123}) \)
Galois group:  $C_2$
Jacobians:  5

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 5 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 323 114019 38284544 12897943299 4346594695283 1464803559740416 493638820954076099 166356282594662950275 56062067226213139367168 18892916655133535897739139

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 323 114019 38284544 12897943299 4346594695283 1464803559740416 493638820954076099 166356282594662950275 56062067226213139367168 18892916655133535897739139

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{337}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-1123}) \).
All geometric endomorphisms are defined over $\F_{337}$.

Base change

This is a primitive isogeny class.

Twists

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