Properties

Label 1.337.a
Base Field $\F_{337}$
Dimension $1$
Ordinary No
$p$-rank $0$
Principally polarizable Yes
Contains a Jacobian Yes

Learn more about

Invariants

Base field:  $\F_{337}$
Dimension:  $1$
L-polynomial:  $1 + 337 x^{2}$
Frobenius angles:  $\pm0.5$
Angle rank:  $0$ (numerical)
Number field:  \(\Q(\sqrt{-337}) \)
Galois group:  $C_2$
Jacobians:  8

This isogeny class is simple and geometrically simple.

Newton polygon

This isogeny class is supersingular.

$p$-rank:  $0$
Slopes:  $[1/2, 1/2]$

Point counts

This isogeny class contains the Jacobians of 8 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 338 114244 38272754 12897690624 4346598285458 1464803698744516 493638820681066034 166356282543723417600 56062067225927988301778 18892916655146425254269764

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 338 114244 38272754 12897690624 4346598285458 1464803698744516 493638820681066034 166356282543723417600 56062067225927988301778 18892916655146425254269764

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{337}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-337}) \).
Endomorphism algebra over $\overline{\F}_{337}$
The base change of $A$ to $\F_{337^{2}}$ is the simple isogeny class 1.113569.zy and its endomorphism algebra is the quaternion algebra over \(\Q\) ramified at $337$ and $\infty$.
All geometric endomorphisms are defined over $\F_{337^{2}}$.

Base change

This is a primitive isogeny class.

Twists

This isogeny class has no twists.