Properties

Label 1.19.ab
Base Field $\F_{19}$
Dimension $1$
Ordinary Yes
$p$-rank $1$
Principally polarizable Yes
Contains a Jacobian Yes

Learn more about

Invariants

Base field:  $\F_{19}$
Dimension:  $1$
L-polynomial:  $1 - x + 19 x^{2}$
Frobenius angles:  $\pm0.463406802480$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-3}) \)
Galois group:  $C_2$
Jacobians:  3

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 3 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 19 399 6916 129675 2474389 47056464 893914831 16983405075 322686721084 6131068282479

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 19 399 6916 129675 2474389 47056464 893914831 16983405075 322686721084 6131068282479

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{19}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}) \).
All geometric endomorphisms are defined over $\F_{19}$.

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon base change
1.19.b$2$1.361.bl
1.19.ah$3$(not in LMFDB)
1.19.i$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
1.19.b$2$1.361.bl
1.19.ah$3$(not in LMFDB)
1.19.i$3$(not in LMFDB)
1.19.ai$6$(not in LMFDB)
1.19.h$6$(not in LMFDB)