Properties

Label 1.27.e
Base field $\F_{3^{3}}$
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:  $\F_{3^{3}}$
Dimension:  $1$
L-polynomial:  $1 + 4 x + 27 x^{2}$
Frobenius angles:  $\pm0.625764130125$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-23}) \)
Galois group:  $C_2$
Jacobians:  $6$
Isomorphism classes:  6

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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $32$ $768$ $19424$ $531456$ $14355872$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $32$ $768$ $19424$ $531456$ $14355872$ $387392256$ $10460278112$ $282430599168$ $7625595261728$ $205891112295168$

Jacobians and polarizations

This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{3^{3}}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-23}) \).

Base change

This is a primitive isogeny class.

Twists

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

TwistExtension degreeCommon base change
1.27.ae$2$1.729.bm