Properties

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

Learn more about

Invariants

Base field:  $\F_{3^{4}}$
Dimension:  $2$
L-polynomial:  $1 - 28 x + 356 x^{2} - 2268 x^{3} + 6561 x^{4}$
Frobenius angles:  $\pm0.172729634987$, $\pm0.253535082407$
Angle rank:  $2$ (numerical)
Number field:  4.0.915712.1
Galois group:  $D_{4}$
Jacobians:  16

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 16 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 4622 42587108 283040783438 1853903518591888 12158314616115510382 79766721320163670059428 523347640026643140373205998 3433683719259797146373768712192 22528399451480173884864760885901678 147808829368216959981224260956276399908

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 54 6490 532590 43067238 3486970574 282430521658 22876792760918 1853020134328574 150094634674331958 12157665455262700090

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{4}}$
The endomorphism algebra of this simple isogeny class is 4.0.915712.1.
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.bc_ns$2$(not in LMFDB)