Properties

Label 2.81.abc_ng
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:  $1 - 28 x + 344 x^{2} - 2268 x^{3} + 6561 x^{4}$
Frobenius angles:  $\pm0.0539942730482$, $\pm0.306978860048$
Angle rank:  $2$ (numerical)
Number field:  4.0.6334720.3
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
$A(\F_{q^r})$ 4610 42421220 282503201810 1853006571051280 12157348072029135250 79766021494260220470500 523347355822296617516069890 3433683764381938628509465067520 22528399625284992067353363690641090 147808829507709580004996979704132310500

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 54 6466 531582 43046406 3486693374 282428043778 22876780337654 1853020158679166 150094635832300182 12157665466736335426

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{4}}$
The endomorphism algebra of this simple isogeny class is 4.0.6334720.3.
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_ng$2$(not in LMFDB)