Properties

Label 2.27.am_dk
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:  $1 - 12 x + 88 x^{2} - 324 x^{3} + 729 x^{4}$
Frobenius angles:  $\pm0.247139293020$, $\pm0.354528994537$
Angle rank:  $2$ (numerical)
Number field:  4.0.295168.1
Galois group:  $D_{4}$
Jacobians:  $12$
Isomorphism classes:  12

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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $482$ $556228$ $396725042$ $283402615824$ $205889234724962$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $16$ $762$ $20152$ $533270$ $14348776$ $387386346$ $10460206192$ $282429401630$ $7625597837392$ $205891125449562$

Jacobians and polarizations

This isogeny class contains the Jacobians of 12 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 4.0.295168.1.

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.m_dk$2$2.729.bg_ccw