Properties

Label 2.81.abd_ny
Base Field $\F_{3^{4}}$
Dimension $2$
Ordinary Yes
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{3^{4}}$
Dimension:  $2$
L-polynomial:  $1 - 29 x + 362 x^{2} - 2349 x^{3} + 6561 x^{4}$
Frobenius angles:  $\pm0.0580440523198$, $\pm0.284000150427$
Angle rank:  $2$ (numerical)
Number field:  4.0.3516652.1
Galois group:  $D_{4}$
Jacobians:  12

This isogeny class is simple and geometrically simple.

Newton polygon

This isogeny class is ordinary.

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

Point counts

This isogeny class contains the Jacobians of 12 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 4546 42286892 282459800200 1853113095980800 12157515710003257106 79766125852891924985600 523347350858005750641704786 3433683697717156530003068083200 22528399573470900792358054980545800 147808829509839558975231622091913144812

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 53 6445 531500 43048881 3486741453 282428413282 22876780120653 1853020122702881 150094635487090700 12157665466911531805

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{4}}$
The endomorphism algebra of this simple isogeny class is 4.0.3516652.1.
All geometric endomorphisms are defined over $\F_{3^{4}}$.

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.bd_ny$2$(not in LMFDB)