Invariants
| Base field: | $\F_{31}$ |
| Dimension: | $1$ |
| L-polynomial: | $1 + 31 x^{2}$ |
| Frobenius angles: | $\pm0.5$ |
| Angle rank: | $0$ (numerical) |
| Number field: | \(\Q(\sqrt{-31}) \) |
| Galois group: | $C_2$ |
| Jacobians: | $6$ |
| Isomorphism classes: | 6 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $32$ | $1024$ | $29792$ | $921600$ | $28629152$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $32$ | $1024$ | $29792$ | $921600$ | $28629152$ | $887563264$ | $27512614112$ | $852889190400$ | $26439622160672$ | $819628344239104$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{31^{2}}$.
Endomorphism algebra over $\F_{31}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-31}) \). |
| The base change of $A$ to $\F_{31^{2}}$ is the simple isogeny class 1.961.ck and its endomorphism algebra is the quaternion algebra over \(\Q\) ramified at $31$ and $\infty$. |
Base change
This is a primitive isogeny class.
Twists
This isogeny class has no twists.