Invariants
| Base field: | $\F_{67}$ |
| Dimension: | $1$ |
| L-polynomial: | $1 + 4 x + 67 x^{2}$ |
| Frobenius angles: | $\pm0.578570930462$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-7}) \) |
| Galois group: | $C_2$ |
| Jacobians: | $10$ |
| Isomorphism classes: | 10 |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2, 3$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $72$ | $4608$ | $300024$ | $20146176$ | $1350194472$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $72$ | $4608$ | $300024$ | $20146176$ | $1350194472$ | $90458436096$ | $6060706742232$ | $406067693395968$ | $27206534658764808$ | $1822837802440647168$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 10 curves (of which 0 are hyperelliptic):
- $y^2=x^3+39 x+39$
- $y^2=x^3+20 x+40$
- $y^2=x^3+13 x+13$
- $y^2=x^3+65 x+63$
- $y^2=x^3+x+2$
- $y^2=x^3+32 x+32$
- $y^2=x^3+41 x+41$
- $y^2=x^3+56 x+45$
- $y^2=x^3+55 x+55$
- $y^2=x^3+34 x+1$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{67}$.
Endomorphism algebra over $\F_{67}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-7}) \). |
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 |
|---|---|---|
| 1.67.ae | $2$ | (not in LMFDB) |