Properties

Label 1.337.ai
Base field $\F_{337}$
Dimension $1$
$p$-rank $1$
Ordinary yes
Supersingular no
Simple yes
Geometrically simple yes
Primitive yes
Principally polarizable yes
Contains a Jacobian yes

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Invariants

Base field:  $\F_{337}$
Dimension:  $1$
L-polynomial:  $1 - 8 x + 337 x^{2}$
Frobenius angles:  $\pm0.430081329241$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-321}) \)
Galois group:  $C_2$
Jacobians:  $20$

This isogeny class is simple and geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $1$
Slopes:  $[0, 1]$

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $330$ $114180$ $38280330$ $12897772800$ $4346594572650$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $330$ $114180$ $38280330$ $12897772800$ $4346594572650$ $1464803641348740$ $493638822085480170$ $166356282574301107200$ $56062067225492955570570$ $18892916655132640311024900$

Jacobians and polarizations

This isogeny class contains the Jacobians of 20 curves (of which all are hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{337}$.

Endomorphism algebra over $\F_{337}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-321}) \).

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.i$2$(not in LMFDB)