Properties

Label 2.199.abx_bmd
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:  $1 - 49 x + 991 x^{2} - 9751 x^{3} + 39601 x^{4}$
Frobenius angles:  $\pm0.0858903897209$, $\pm0.218783630545$
Angle rank:  $2$ (numerical)
Number field:  4.0.24389.1
Galois group:  $C_4$
Jacobians:  37

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 37 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 30793 1551751649 62094192367879 2459430349031718125 97393934325151794733648 3856887607575479705627088641 152736583352900232055591017099163 6048521412896430746605940610917163125 239527496513711152073095485707223106154029 9485528389648780162019162276841071747212571904

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 151 39183 7879375 1568275011 312080424396 62103850106331 12358664326729669 2459374191072864531 489415464108498577825 97393677359740276704478

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{199}$
The endomorphism algebra of this simple isogeny class is 4.0.24389.1.
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.bx_bmd$2$(not in LMFDB)