Invariants
| Base field: | $\F_{61}$ |
| Dimension: | $1$ |
| L-polynomial: | $1 - 12 x + 61 x^{2}$ |
| Frobenius angles: | $\pm0.221142061624$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-1}) \) |
| Galois group: | $C_2$ |
| Jacobians: | $3$ |
| Isomorphism classes: | 3 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $50$ | $3700$ | $227450$ | $13852800$ | $844651250$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $50$ | $3700$ | $227450$ | $13852800$ | $844651250$ | $51520609300$ | $3142742303450$ | $191707292275200$ | $11694145876656050$ | $713342910332792500$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 3 curves (of which 0 are hyperelliptic):
- $y^2=x^3+x+1$
- $y^2=x^3+6 x$
- $y^2=x^3+44 x+27$
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.m | $2$ | (not in LMFDB) |
| 1.61.ak | $4$ | (not in LMFDB) |
| 1.61.k | $4$ | (not in LMFDB) |