Invariants
| Base field: | $\F_{41}$ |
| Dimension: | $1$ |
| L-polynomial: | $1 + 41 x^{2}$ |
| Frobenius angles: | $\pm0.5$ |
| Angle rank: | $0$ (numerical) |
| Number field: | \(\Q(\sqrt{-41}) \) |
| Galois group: | $C_2$ |
| Jacobians: | $8$ |
| Isomorphism classes: | 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $42$ | $1764$ | $68922$ | $2822400$ | $115856202$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $42$ | $1764$ | $68922$ | $2822400$ | $115856202$ | $4750242084$ | $194754273882$ | $7984919577600$ | $327381934393962$ | $13422659541864804$ |
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 $\F_{41^{2}}$.
Endomorphism algebra over $\F_{41}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-41}) \). |
| The base change of $A$ to $\F_{41^{2}}$ is the simple isogeny class 1.1681.de and its endomorphism algebra is the quaternion algebra over \(\Q\) ramified at $41$ and $\infty$. |
Base change
This is a primitive isogeny class.
Twists
This isogeny class has no twists.