Properties

Label 1.361.c
Base Field $\F_{19^{2}}$
Dimension $1$
Ordinary Yes
$p$-rank $1$
Principally polarizable Yes
Contains a Jacobian Yes

Learn more about

Invariants

Base field:  $\F_{19^{2}}$
Dimension:  $1$
L-polynomial:  $1 + 2 x + 361 x^{2}$
Frobenius angles:  $\pm0.516760896166$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-10}) \)
Galois group:  $C_2$
Jacobians:  30

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 30 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 364 131040 47043724 16983308160 6131067546604 2213315008500960 799006685138757004 288441413536623459840 104127350298205766838124 37589973457556559315276000

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 364 131040 47043724 16983308160 6131067546604 2213315008500960 799006685138757004 288441413536623459840 104127350298205766838124 37589973457556559315276000

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{19^{2}}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-10}) \).
All geometric endomorphisms are defined over $\F_{19^{2}}$.

Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{19^{2}}$.

SubfieldPrimitive Model
$\F_{19}$1.19.ag
$\F_{19}$1.19.g

Twists

Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
1.361.ac$2$(not in LMFDB)