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, primitive, not ordinary, and supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
This isogeny class is supersingular.
| $p$-rank: | $0$ |
| Slopes: | $[1/2, 1/2]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $338$ | $114244$ | $38272754$ | $12897690624$ | $4346598285458$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $338$ | $114244$ | $38272754$ | $12897690624$ | $4346598285458$ | $1464803698744516$ | $493638820681066034$ | $166356282543723417600$ | $56062067225927988301778$ | $18892916655146425254269764$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{337^{2}}$.
Endomorphism algebra over $\F_{337}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-337}) \). |
| 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 $\infty$. |
Base change
This is a primitive isogeny class.
Twists
This isogeny class has no twists.