Properties

Label 2.199.abx_bmg
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:  $1 - 49 x + 994 x^{2} - 9751 x^{3} + 39601 x^{4}$
Frobenius angles:  $\pm0.109464524120$, $\pm0.207308250412$
Angle rank:  $2$ (numerical)
Number field:  4.0.1912313.2
Galois group:  $D_{4}$
Jacobians:  24

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 24 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 30796 1551995216 62097671921392 2459456857100257088 97394075393379673594516 3856888185376959360802091264 152736585224623379978132958633988 6048521417527256449456481434553046272 239527496520758674706290324384242643627888 9485528389643302343976287503677148608831602256

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 151 39189 7879816 1568291913 312080876421 62103859410126 12358664478179947 2459374192955793009 489415464122898455128 97393677359684032621789

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{199}$
The endomorphism algebra of this simple isogeny class is 4.0.1912313.2.
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.bx_bmg$2$(not in LMFDB)