Properties

Label 2.81.abe_ov
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:  $1 - 30 x + 385 x^{2} - 2430 x^{3} + 6561 x^{4}$
Frobenius angles:  $\pm0.134614680791$, $\pm0.227750444521$
Angle rank:  $2$ (numerical)
Number field:  4.0.486976.1
Galois group:  $D_{4}$
Jacobians:  12

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 12 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 4487 42209209 282621574172 1853631300064761 12158251775368239527 79766812365732093948304 523347784423933582361773847 3433683839956882633574097395625 22528399521607965820420932770913692 147808829399195117831955119514368883529

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 52 6432 531802 43060916 3486952552 282430844022 22876799072872 1853020199463908 150094635141555802 12157665457810735152

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{4}}$
The endomorphism algebra of this simple isogeny class is 4.0.486976.1.
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.be_ov$2$(not in LMFDB)