Properties

Label 2.81.abf_pk
Base field $\F_{3^{4}}$
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:  $\F_{3^{4}}$
Dimension:  $2$
L-polynomial:  $( 1 - 17 x + 81 x^{2} )( 1 - 14 x + 81 x^{2} )$
  $1 - 31 x + 400 x^{2} - 2511 x^{3} + 6561 x^{4}$
Frobenius angles:  $\pm0.106600758076$, $\pm0.216346895939$
Angle rank:  $2$ (numerical)
Jacobians:  $14$
Isomorphism classes:  70

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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $4420$ $42007680$ $282364186000$ $1853405894545920$ $12158106367827530500$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $51$ $6401$ $531318$ $43055681$ $3486910851$ $282430617758$ $22876798704531$ $1853020208811521$ $150094635301373718$ $12157665459807085601$

Jacobians and polarizations

This isogeny class contains the Jacobians of 14 curves (of which all are hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{3^{4}}$.

Endomorphism algebra over $\F_{3^{4}}$
The isogeny class factors as 1.81.ar $\times$ 1.81.ao 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
2.81.ad_acy$2$(not in LMFDB)
2.81.d_acy$2$(not in LMFDB)
2.81.bf_pk$2$(not in LMFDB)