Properties

Label 1.419.aw
Base Field $\F_{419}$
Dimension $1$
Ordinary Yes
$p$-rank $1$
Principally polarizable Yes
Contains a Jacobian Yes

Learn more about

Invariants

Base field:  $\F_{419}$
Dimension:  $1$
L-polynomial:  $1 - 22 x + 419 x^{2}$
Frobenius angles:  $\pm0.319412152670$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-298}) \)
Galois group:  $C_2$
Jacobians:  6

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 6 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 398 175916 73577066 30821890528 12914275360318 5411082137999564 2267243473133243002 949975016184390987648 398039531777960704031534 166778563814498439804450476

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 398 175916 73577066 30821890528 12914275360318 5411082137999564 2267243473133243002 949975016184390987648 398039531777960704031534 166778563814498439804450476

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{419}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-298}) \).
All geometric endomorphisms are defined over $\F_{419}$.

Base change

This is a primitive isogeny class.

Twists

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