Properties

Label 2.81.abb_mn
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:  $1 - 27 x + 325 x^{2} - 2187 x^{3} + 6561 x^{4}$
Frobenius angles:  $\pm0.0356095094810$, $\pm0.331030938454$
Angle rank:  $2$ (numerical)
Number field:  4.0.5768917.1
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
$A(\F_{q^r})$ 4673 42528973 282471639473 1852807413525237 12157123583429758208 79765908105094472487037 523347359059024270581661337 3433683792140010132980315011173 22528399598825801686274794000126457 147808829424895571165077143069102542848

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 55 6483 531523 43041779 3486628990 282427642299 22876780479139 1853020173659075 150094635656016799 12157665459924665118

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{4}}$
The endomorphism algebra of this simple isogeny class is 4.0.5768917.1.
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.bb_mn$2$(not in LMFDB)