Properties

Label 2.81.abc_nm
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:  $1 - 28 x + 350 x^{2} - 2268 x^{3} + 6561 x^{4}$
Frobenius angles:  $\pm0.115477597890$, $\pm0.286871664488$
Angle rank:  $2$ (numerical)
Number field:  4.0.130048.1
Galois group:  $D_{4}$
Jacobians:  28

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 28 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 4616 42504128 282771954824 1853458127813632 12157848913418949256 79766421796524634715072 523347594614176631679881224 3433683879480993415489966030848 22528399689928369358706582110789384 147808829566974499476487216073818893248

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 54 6478 532086 43056894 3486837014 282429461134 22876790775830 1853020220793470 150094636262984310 12157665471611031118

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{4}}$
The endomorphism algebra of this simple isogeny class is 4.0.130048.1.
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_nm$2$(not in LMFDB)