Properties

Label 1.361.abe
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:  $1 - 30 x + 361 x^{2}$
Frobenius angles:  $\pm0.210353590899$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-34}) \)
Galois group:  $C_2$
Jacobians:  12

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 12 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 332 130144 47051372 16983792000 6131071144652 2213314983017824 799006685937281132 288441413549166528000 104127350297301865021772 37589973457534339022948704

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 332 130144 47051372 16983792000 6131071144652 2213314983017824 799006685937281132 288441413549166528000 104127350297301865021772 37589973457534339022948704

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{19^{2}}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-34}) \).
All geometric endomorphisms are defined over $\F_{19^{2}}$.

Base change

This is a primitive isogeny class.

Twists

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