Properties

Label 1.331.aj
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

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Invariants

Base field:  $\F_{331}$
Dimension:  $1$
L-polynomial:  $1 - 9 x + 331 x^{2}$
Frobenius angles:  $\pm0.420442761943$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-1243}) \)
Galois group:  $C_2$
Jacobians:  $4$

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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $323$ $110143$ $36272900$ $12003494283$ $3973192027853$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $323$ $110143$ $36272900$ $12003494283$ $3973192027853$ $1315127818483600$ $435307307509263743$ $144086718365732452563$ $47692703775414252475100$ $15786284949768293868390703$

Jacobians and polarizations

This isogeny class contains the Jacobians of 4 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 \(\Q(\sqrt{-1243}) \).

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.j$2$(not in LMFDB)