Properties

Label 2.9.ag_u
Base field $\F_{3^{2}}$
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:  $\F_{3^{2}}$
Dimension:  $2$
L-polynomial:  $1 - 6 x + 20 x^{2} - 54 x^{3} + 81 x^{4}$
Frobenius angles:  $\pm0.109926884584$, $\pm0.481195587521$
Angle rank:  $2$ (numerical)
Number field:  4.0.116032.1
Galois group:  $D_{4}$
Jacobians:  $8$
Isomorphism classes:  8

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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $42$ $6804$ $519498$ $41830992$ $3482994522$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $4$ $86$ $712$ $6374$ $58984$ $533510$ $4788004$ $43047230$ $387439876$ $3486994886$

Jacobians and polarizations

This isogeny class contains the Jacobians of 8 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 endomorphism algebra of this simple isogeny class is 4.0.116032.1.

Base change

This is a primitive isogeny class.

Twists

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

TwistExtension degreeCommon base change
2.9.g_u$2$2.81.e_adi