Properties

Label 2.199.aby_bnb
Base Field $\F_{199}$
Dimension $2$
Ordinary Yes
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{199}$
Dimension:  $2$
L-polynomial:  $1 - 50 x + 1015 x^{2} - 9950 x^{3} + 39601 x^{4}$
Frobenius angles:  $\pm0.0526487022840$, $\pm0.212225479551$
Angle rank:  $2$ (numerical)
Number field:  4.0.420416.2
Galois group:  $D_{4}$
Jacobians:  21

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 21 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 30617 1549740689 62083355259632 2459386664062026041 97393783544573762510577 3856887121558863409012615424 152736581786635803247227381802097 6048521407696274913078118348591075625 239527496496312327480262728701013024428912 9485528389593066688979499969476166353788372929

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 150 39132 7878000 1568247156 312079941250 62103842280462 12358664199995550 2459374188958442148 489415464072948363600 97393677359168232660652

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{199}$
The endomorphism algebra of this simple isogeny class is 4.0.420416.2.
All geometric endomorphisms are defined over $\F_{199}$.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.199.by_bnb$2$(not in LMFDB)