Properties

Label 2.27.as_fe
Base field $\F_{3^{3}}$
Dimension $2$
$p$-rank $2$
Ordinary yes
Supersingular no
Simple no
Geometrically simple no
Primitive no
Principally polarizable yes
Contains a Jacobian yes

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Invariants

Base field:  $\F_{3^{3}}$
Dimension:  $2$
L-polynomial:  $( 1 - 10 x + 27 x^{2} )( 1 - 8 x + 27 x^{2} )$
  $1 - 18 x + 134 x^{2} - 486 x^{3} + 729 x^{4}$
Frobenius angles:  $\pm0.0877398280459$, $\pm0.220355751984$
Angle rank:  $2$ (numerical)
Jacobians:  $6$
Isomorphism classes:  16

This isogeny class is not simple, not 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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $360$ $492480$ $386371080$ $282801715200$ $205973541793800$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $10$ $674$ $19630$ $532142$ $14354650$ $387444626$ $10460397790$ $282429385118$ $7625596330090$ $205891135662914$

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 isogeny class factors as 1.27.ak $\times$ 1.27.ai and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

Base change

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

SubfieldPrimitive Model
$\F_{3}$2.3.d_i

Twists

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

TwistExtension degreeCommon base change
2.27.ac_aba$2$2.729.ace_cvu
2.27.c_aba$2$2.729.ace_cvu
2.27.s_fe$2$2.729.ace_cvu