Properties

Label 1.331.as
Base field $\F_{331}$
Dimension $1$
$p$-rank $1$
Ordinary yes
Supersingular no
Simple yes
Geometrically simple yes
Primitive yes
Principally polarizable yes
Contains a Jacobian yes

Related objects

Downloads

Learn more

Invariants

Base field:  $\F_{331}$
Dimension:  $1$
L-polynomial:  $1 - 18 x + 331 x^{2}$
Frobenius angles:  $\pm0.335283534355$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-10}) \)
Galois group:  $C_2$
Jacobians:  $12$

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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $314$ $109900$ $36276734$ $12003717600$ $3973193712554$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $314$ $109900$ $36276734$ $12003717600$ $3973193712554$ $1315127740845100$ $435307305600557774$ $144086718368760854400$ $47692703776190360242394$ $15786284949778201421447500$

Jacobians and polarizations

This isogeny class contains the Jacobians of 12 curves (of which all are hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{331}$.

Endomorphism algebra over $\F_{331}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-10}) \).

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

TwistExtension degreeCommon base change
1.331.s$2$(not in LMFDB)