Properties

Label 2.3.ad_i
Base field $\F_{3}$
Dimension $2$
$p$-rank $2$
Ordinary yes
Supersingular no
Simple no
Geometrically simple no
Primitive yes
Principally polarizable yes
Contains a Jacobian no

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Invariants

Base field:  $\F_{3}$
Dimension:  $2$
L-polynomial:  $( 1 - 2 x + 3 x^{2} )( 1 - x + 3 x^{2} )$
  $1 - 3x + 8x^{2} - 9x^{3} + 9x^{4}$
Frobenius angles:  $\pm0.304086723985$, $\pm0.406785250661$
Angle rank:  $2$ (numerical)
Jacobians:  0

This isogeny class is not simple.

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $2$
Slopes:  $[0, 0, 1, 1]$

Point counts

This isogeny class is principally polarizable, but does not contain a Jacobian.

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $6$ $180$ $1368$ $7200$ $51546$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $1$ $17$ $46$ $89$ $211$ $674$ $2185$ $6641$ $19738$ $59057$

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
The isogeny class factors as 1.3.ac $\times$ 1.3.ab and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
All geometric endomorphisms are defined over $\F_{3}$.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.
TwistExtension degreeCommon base change
2.3.ab_e$2$2.9.h_bc
2.3.b_e$2$2.9.h_bc
2.3.d_i$2$2.9.h_bc