Invariants
| Base field: | $\F_{61}$ |
| Dimension: | $1$ |
| L-polynomial: | $1 + 10 x + 61 x^{2}$ |
| Frobenius angles: | $\pm0.721142061624$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-1}) \) |
| Galois group: | $C_2$ |
| Jacobians: | $8$ |
| Isomorphism classes: | 8 |
This isogeny class is simple and geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
This isogeny class is ordinary.
| $p$-rank: | $1$ |
| Slopes: | $[0, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $72$ | $3744$ | $226152$ | $13852800$ | $844577352$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $72$ | $3744$ | $226152$ | $13852800$ | $844577352$ | $51520139424$ | $3142746341352$ | $191707292275200$ | $11694146086229832$ | $713342912992972704$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which 0 are hyperelliptic):
- $y^2=x^3+2 x+2$
- $y^2=x^3+59 x+57$
- $y^2=x^3+16 x+16$
- $y^2=x^3+3 x$
- $y^2=x^3+7 x+14$
- $y^2=x^3+31 x+31$
- $y^2=x^3+47 x+47$
- $y^2=x^3+55 x+49$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{61}$.
Endomorphism algebra over $\F_{61}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-1}) \). |
Base change
This is a primitive isogeny class.
Twists
Below is a list of all twists of this isogeny class.
| Twist | Extension degree | Common base change |
|---|---|---|
| 1.61.ak | $2$ | (not in LMFDB) |
| 1.61.am | $4$ | (not in LMFDB) |
| 1.61.m | $4$ | (not in LMFDB) |