Properties

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

Learn more about

Invariants

Base field:  $\F_{199}$
Dimension:  $2$
L-polynomial:  $1 - 50 x + 1016 x^{2} - 9950 x^{3} + 39601 x^{4}$
Frobenius angles:  $\pm0.0639639324574$, $\pm0.208870266773$
Angle rank:  $2$ (numerical)
Number field:  4.0.90944.1
Galois group:  $D_{4}$
Jacobians:  36

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 36 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 30618 1549821924 62084538832050 2459395976883917904 97393835662033816874178 3856887352730251658934202500 152736582642442407857243727078858 6048521410414619301857434046712812544 239527496503855464536575353395231615133450 9485528389611750908959447800007415269502175684

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 150 39134 7878150 1568253094 312080108250 62103846002798 12358664269243050 2459374190063741374 489415464088360906950 97393677359360074887854

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{199}$
The endomorphism algebra of this simple isogeny class is 4.0.90944.1.
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_bnc$2$(not in LMFDB)