Properties

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

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Invariants

Base field:  $\F_{3^{3}}$
Dimension:  $2$
L-polynomial:  $( 1 - 10 x + 27 x^{2} )( 1 - 7 x + 27 x^{2} )$
  $1 - 17 x + 124 x^{2} - 459 x^{3} + 729 x^{4}$
Frobenius angles:  $\pm0.0877398280459$, $\pm0.264757707515$
Angle rank:  $2$ (numerical)
Jacobians:  $3$
Isomorphism classes:  15

This isogeny class is not 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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $378$ $502740$ $388086552$ $282841524000$ $205927500801078$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $11$ $689$ $19718$ $532217$ $14351441$ $387412946$ $10460241803$ $282429175793$ $7625599857746$ $205891171523489$

Jacobians and polarizations

This isogeny class contains the Jacobians of 3 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.ah 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 is a primitive isogeny class.

Twists

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

TwistExtension degreeCommon base change
2.27.ad_aq$2$2.729.abp_bvg
2.27.d_aq$2$2.729.abp_bvg
2.27.r_eu$2$2.729.abp_bvg