Properties

Label 1.9.c
Base field $\F_{3^{2}}$
Dimension $1$
$p$-rank $1$
Ordinary yes
Supersingular no
Simple yes
Geometrically simple yes
Primitive no
Principally polarizable yes
Contains a Jacobian yes

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Invariants

Base field:  $\F_{3^{2}}$
Dimension:  $1$
L-polynomial:  $1 + 2 x + 9 x^{2}$
Frobenius angles:  $\pm0.608173447969$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-2}) \)
Galois group:  $C_2$
Jacobians:  $3$
Isomorphism classes:  3

This isogeny class is simple and geometrically simple, not 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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $12$ $96$ $684$ $6528$ $59532$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $12$ $96$ $684$ $6528$ $59532$ $530784$ $4779948$ $43058688$ $387423756$ $3486670176$

Jacobians and polarizations

This isogeny class contains the Jacobians of 3 curves (of which all are hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{3^{2}}$.

Endomorphism algebra over $\F_{3^{2}}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-2}) \).

Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{3^{2}}$.

SubfieldPrimitive Model
$\F_{3}$1.3.ac
$\F_{3}$1.3.c

Twists

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

TwistExtension degreeCommon base change
1.9.ac$2$1.81.o