Properties

Label 2.199.aby_bne
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:  $1 - 50 x + 1018 x^{2} - 9950 x^{3} + 39601 x^{4}$
Frobenius angles:  $\pm0.0840302640260$, $\pm0.201172616085$
Angle rank:  $2$ (numerical)
Number field:  4.0.376400.1
Galois group:  $D_{4}$
Jacobians:  54

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 54 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 30620 1549984400 62086905995180 2459414583733126400 97393939428854323975500 3856887808961276372867632400 152736584298256252785492947433980 6048521415452027726835637992024166400 239527496516564876640012642344705882740220 9485528389636924418573186328242558172350250000

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 150 39138 7878450 1568264958 312080440750 62103853349058 12358664403223050 2459374192111989438 489415464114329461350 97393677359618546590498

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{199}$
The endomorphism algebra of this simple isogeny class is 4.0.376400.1.
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_bne$2$(not in LMFDB)