Invariants
| Base field: | $\F_{7}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 4 x + 12 x^{2} - 28 x^{3} + 49 x^{4}$ |
| Frobenius angles: | $\pm0.182041207691$, $\pm0.527071640754$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-18 +4 \sqrt{6}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $4$ |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $30$ | $2820$ | $116910$ | $5696400$ | $288519150$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $4$ | $58$ | $340$ | $2374$ | $17164$ | $118906$ | $823708$ | $5761726$ | $40357060$ | $282469018$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 4 curves (of which all are hyperelliptic):
- $y^2=6 x^6+2 x^5+5 x^4+6 x^3+4 x^2+x+6$
- $y^2=5 x^6+2 x^5+6 x^3+4 x^2+3 x$
- $y^2=4 x^5+3 x^4+4 x^3+6 x^2+5 x+6$
- $y^2=3 x^6+4 x^5+4 x^4+5 x^3+x^2+6 x+3$
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 \(\Q(\sqrt{-18 +4 \sqrt{6}})\). |
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.e_m | $2$ | 2.49.i_s |