Invariants
| Base field: | $\F_{163}$ |
| Dimension: | $1$ |
| L-polynomial: | $1 + 163 x^{2}$ |
| Frobenius angles: | $\pm0.5$ |
| Angle rank: | $0$ (numerical) |
| Number field: | \(\Q(\sqrt{-163}) \) |
| Galois group: | $C_2$ |
| Jacobians: | $4$ |
| Isomorphism classes: | 4 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $164$ | $26896$ | $4330748$ | $705858624$ | $115063617044$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $164$ | $26896$ | $4330748$ | $705858624$ | $115063617044$ | $18755378239504$ | $3057125241215468$ | $498311412906297600$ | $81224760533853742724$ | $13239635967248287297936$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 4 curves (of which 0 are hyperelliptic):
- $y^2=x^3+56 x+112$
- $y^2=x^3+2 x$
- $y^2=x^3+x$
- $y^2=x^3+14 x+14$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{163^{2}}$.
Endomorphism algebra over $\F_{163}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-163}) \). |
| The base change of $A$ to $\F_{163^{2}}$ is the simple isogeny class 1.26569.mo and its endomorphism algebra is the quaternion algebra over \(\Q\) ramified at $163$ and $\infty$. |
Base change
This is a primitive isogeny class.
Twists
This isogeny class has no twists.