Properties

Label 2.243.ace_bwr
Base field $\F_{3^{5}}$
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:  $\F_{3^{5}}$
Dimension:  $2$
L-polynomial:  $1 - 56 x + 1265 x^{2} - 13608 x^{3} + 59049 x^{4}$
Frobenius angles:  $\pm0.0783982229649$, $\pm0.190397219869$
Angle rank:  $2$ (numerical)
Number field:  4.0.445225.1
Galois group:  $D_{4}$

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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $46651$ $3451194329$ $205834903662784$ $12157739442976229641$ $717899010627359245105091$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $188$ $58444$ $14344988$ $3486805620$ $847289816748$ $205891155303238$ $50031545391647700$ $12157665461262443684$ $2954312706548675193764$ $717897987691553646133564$

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 endomorphism algebra of this simple isogeny class is 4.0.445225.1.

Base change

This is a primitive isogeny class.

Twists

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

TwistExtension degreeCommon base change
2.243.ce_bwr$2$(not in LMFDB)