Properties

Label 2.9.ag_x
Base field $\F_{3^{2}}$
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:  $\F_{3^{2}}$
Dimension:  $2$
L-polynomial:  $( 1 - 5 x + 9 x^{2} )( 1 - x + 9 x^{2} )$
  $1 - 6 x + 23 x^{2} - 54 x^{3} + 81 x^{4}$
Frobenius angles:  $\pm0.186429498677$, $\pm0.446699620962$
Angle rank:  $2$ (numerical)
Jacobians:  $12$
Isomorphism classes:  20

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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $45$ $7425$ $559440$ $42953625$ $3493462725$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $4$ $92$ $766$ $6548$ $59164$ $533582$ $4789516$ $43044068$ $387360334$ $3486664652$

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 $\F_{3^{2}}$.

Endomorphism algebra over $\F_{3^{2}}$
The isogeny class factors as 1.9.af $\times$ 1.9.ab 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
2.9.ae_n$2$2.81.k_br
2.9.e_n$2$2.81.k_br
2.9.g_x$2$2.81.k_br