Properties

Label 2.27.an_df
Base field $\F_{3^{3}}$
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:  $\F_{3^{3}}$
Dimension:  $2$
L-polynomial:  $1 - 13 x + 83 x^{2} - 351 x^{3} + 729 x^{4}$
Frobenius angles:  $\pm0.0702760081003$, $\pm0.411256475822$
Angle rank:  $2$ (numerical)
Number field:  4.0.1452253.1
Galois group:  $D_{4}$
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:  $2$
Slopes:  $[0, 0, 1, 1]$

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $449$ $528473$ $387157883$ $281598423469$ $205735242292624$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $15$ $727$ $19671$ $529875$ $14338040$ $387406891$ $10460537921$ $282430394195$ $7625596453269$ $205891122241462$

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 4.0.1452253.1.

Base change

This is a primitive isogeny class.

Twists

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

TwistExtension degreeCommon base change
2.27.n_df$2$2.729.ad_abdz