Properties

Label 1.199.ag
Base Field $\F_{199}$
Dimension $1$
Ordinary Yes
$p$-rank $1$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{199}$
Dimension:  $1$
L-polynomial:  $1 - 6 x + 199 x^{2}$
Frobenius angles:  $\pm0.431785908339$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-190}) \)
Galois group:  $C_2$
Jacobians:  4

This isogeny class is simple and geometrically simple.

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $1$
Slopes:  $[0, 1]$

Point counts

This isogeny class contains the Jacobians of 4 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 194 39964 7883966 1568187360 312078620114 62103845030044 12358664500945166 2459374192002003840 489415464077628904994 97393677359357063655004

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 194 39964 7883966 1568187360 312078620114 62103845030044 12358664500945166 2459374192002003840 489415464077628904994 97393677359357063655004

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{199}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-190}) \).
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
1.199.g$2$(not in LMFDB)