Invariants
| Base field: | $\F_{61}$ |
| Dimension: | $1$ |
| L-polynomial: | $1 + 61 x^{2}$ |
| Frobenius angles: | $\pm0.5$ |
| Angle rank: | $0$ (numerical) |
| Number field: | \(\Q(\sqrt{-61}) \) |
| Galois group: | $C_2$ |
| Jacobians: | $6$ |
| Isomorphism classes: | 6 |
| Cyclic group of points: | yes |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $62$ | $3844$ | $226982$ | $13838400$ | $844596302$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $62$ | $3844$ | $226982$ | $13838400$ | $844596302$ | $51520828324$ | $3142742836022$ | $191707285305600$ | $11694146092834142$ | $713342913352075204$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 6 curves (of which 0 are hyperelliptic):
- $y^2=x^3+11 x+22$
- $y^2=x^3+18 x+18$
- $y^2=x^3+4 x+4$
- $y^2=x^3+50 x+50$
- $y^2=x^3+16 x+32$
- $y^2=x^3+17 x+34$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{61^{2}}$.
Endomorphism algebra over $\F_{61}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-61}) \). |
| The base change of $A$ to $\F_{61^{2}}$ is the simple isogeny class 1.3721.es and its endomorphism algebra is the quaternion algebra over \(\Q\) ramified at $61$ and $\infty$. |
Base change
This is a primitive isogeny class.
Twists
This isogeny class has no twists.