Properties

Label 2.199.aby_bnd
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:  $1 - 50 x + 1017 x^{2} - 9950 x^{3} + 39601 x^{4}$
Frobenius angles:  $\pm0.0742443472140$, $\pm0.205213560661$
Angle rank:  $2$ (numerical)
Number field:  4.0.7041600.5
Galois group:  $D_{4}$
Jacobians:  16

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 16 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 30619 1549903161 62085722410564 2459405283440943225 97393887623460655401379 3856887581864140880007438096 152736583479640573289442679906339 6048521412999713489092853046270045225 239527496510604346953145405586270295887044 9485528389626350956946550953863407370392601641

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 150 39136 7878300 1568259028 312080274750 62103849692326 12358664336984850 2459374191114860068 489415464102150586500 97393677359509982442256

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{199}$
The endomorphism algebra of this simple isogeny class is 4.0.7041600.5.
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_bnd$2$(not in LMFDB)