Properties

Label 2.81.abc_nr
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:  $1 - 28 x + 355 x^{2} - 2268 x^{3} + 6561 x^{4}$
Frobenius angles:  $\pm0.161515673273$, $\pm0.261305008072$
Angle rank:  $2$ (numerical)
Number field:  4.0.1911312.2
Galois group:  $D_{4}$
Jacobians:  24

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 24 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 4621 42573273 282995973364 1853829714911577 12158239439137856581 79766678366986982306832 523347645640407947554716949 3433683764090739377272547490153 22528399509587350399441439059372084 147808829413820480777577630855396809673

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 54 6488 532506 43065524 3486949014 282430369574 22876793006310 1853020158522020 150094635061468890 12157665459013709768

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{4}}$
The endomorphism algebra of this simple isogeny class is 4.0.1911312.2.
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_nr$2$(not in LMFDB)