Properties

Label 2.243.aci_cbh
Base field $\F_{3^{5}}$
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:  $\F_{3^{5}}$
Dimension:  $2$
L-polynomial:  $( 1 - 31 x + 243 x^{2} )( 1 - 29 x + 243 x^{2} )$
  $1 - 60 x + 1385 x^{2} - 14580 x^{3} + 59049 x^{4}$
Frobenius angles:  $\pm0.0339262533067$, $\pm0.119654564389$
Angle rank:  $2$ (numerical)

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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $45795$ $3438059625$ $205741381552560$ $12157262903676515625$ $717897118120360293788475$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $184$ $58220$ $14338468$ $3486668948$ $847287583144$ $205891127215910$ $50031545160853048$ $12157665461379173348$ $2954312706593971455004$ $717897987692414451871100$

Jacobians and polarizations

This isogeny class contains a Jacobian, and hence is principally polarizable.

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{3^{5}}$.

Endomorphism algebra over $\F_{3^{5}}$
The isogeny class factors as 1.243.abf $\times$ 1.243.abd 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
2.243.ac_apx$2$(not in LMFDB)
2.243.c_apx$2$(not in LMFDB)
2.243.ci_cbh$2$(not in LMFDB)