Properties

Label 2.3.ac_e
Base field $\F_{3}$
Dimension $2$
$p$-rank $2$
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:  $\F_{3}$
Dimension:  $2$
L-polynomial:  $1 - 2 x + 4 x^{2} - 6 x^{3} + 9 x^{4}$
Frobenius angles:  $\pm0.210767374595$, $\pm0.567777800232$
Angle rank:  $2$ (numerical)
Number field:  4.0.7488.1
Galois group:  $D_{4}$
Jacobians:  2

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:  $2$
Slopes:  $[0, 0, 1, 1]$

Point counts

This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $6$ $132$ $702$ $6864$ $74526$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $2$ $14$ $26$ $86$ $302$ $782$ $2102$ $6494$ $19682$ $58334$

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
The endomorphism algebra of this simple isogeny class is 4.0.7488.1.
All geometric endomorphisms are defined over $\F_{3}$.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.
TwistExtension degreeCommon base change
2.3.c_e$2$2.9.e_k