Properties

Label 2.81.abd_nx
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:  $1 - 29 x + 361 x^{2} - 2349 x^{3} + 6561 x^{4}$
Frobenius angles:  $\pm0.0405552772615$, $\pm0.287450300601$
Angle rank:  $2$ (numerical)
Number field:  4.0.26125.1
Galois group:  $D_{4}$
Jacobians:  20

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 20 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 4545 42273045 282413396745 1853030515239045 12157415098767450000 79766031668791568059845 523347276925733859867609945 3433683642766367124221468155845 22528399528490497335275231503415745 147808829467835157825336903404659200000

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 53 6443 531413 43046963 3486712598 282428079803 22876776888893 1853020093048163 150094635187410413 12157665463456559198

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{4}}$
The endomorphism algebra of this simple isogeny class is 4.0.26125.1.
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_nx$2$(not in LMFDB)