Properties

Label 2.81.abf_pi
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:  $1 - 31 x + 398 x^{2} - 2511 x^{3} + 6561 x^{4}$
Frobenius angles:  $\pm0.0704000604239$, $\pm0.231694314237$
Angle rank:  $2$ (numerical)
Number field:  4.0.646204.2
Galois group:  $D_{4}$
Jacobians:  8

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 8 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 4418 41979836 282264888992 1853212286251456 12157842620564660098 79766472156290208383744 523347546330792038254761202 3433683708815352136166108047104 22528399480178107852003180754079392 147808829418923839671288258587540090876

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 51 6397 531132 43051185 3486835211 282429639442 22876788665243 1853020128692129 150094634865530892 12157665459433474477

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{4}}$
The endomorphism algebra of this simple isogeny class is 4.0.646204.2.
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.bf_pi$2$(not in LMFDB)