Properties

Label 2.243.acf_bxy
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 no

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Invariants

Base field:  $\F_{3^{5}}$
Dimension:  $2$
L-polynomial:  $( 1 - 29 x + 243 x^{2} )( 1 - 28 x + 243 x^{2} )$
  $1 - 57 x + 1298 x^{2} - 13851 x^{3} + 59049 x^{4}$
Frobenius angles:  $\pm0.119654564389$, $\pm0.144947286894$
Angle rank:  $2$ (numerical)

This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable.

Newton polygon

This isogeny class is ordinary.

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

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $46440$ $3448448640$ $205822463222880$ $12157739967920601600$ $717899473696939086742200$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $187$ $58397$ $14344120$ $3486805769$ $847290363277$ $205891176569174$ $50031545936253271$ $12157665472083090641$ $2954312706719024345800$ $717897987693502120524557$

Jacobians and polarizations

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

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.abd $\times$ 1.243.abc 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.ab_amo$2$(not in LMFDB)
2.243.b_amo$2$(not in LMFDB)
2.243.cf_bxy$2$(not in LMFDB)