Invariants
| Base field: | $\F_{7}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - x - x^{2} - 7 x^{3} + 49 x^{4}$ |
| Frobenius angles: | $\pm0.186914372711$, $\pm0.722521724645$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-50 +2 \sqrt{61}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $2$ |
| Isomorphism classes: | 2 |
| Cyclic group of points: | yes |
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: | $2$ |
| Slopes: | $[0, 0, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $41$ | $2337$ | $109511$ | $6162669$ | $285242576$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $7$ | $47$ | $319$ | $2563$ | $16972$ | $117947$ | $826357$ | $5761171$ | $40346833$ | $282469982$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 2 curves (of which all are hyperelliptic):
- $y^2=6 x^6+5 x^5+5 x^4+5 x^3+4 x^2+2 x+6$
- $y^2=x^6+x^5+3 x^2+2 x+6$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{7}$.
Endomorphism algebra over $\F_{7}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-50 +2 \sqrt{61}})\). |
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 |
|---|---|---|
| 2.7.b_ab | $2$ | 2.49.ad_dh |