Invariants
| Base field: | $\F_{7}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - x + 13 x^{2} - 7 x^{3} + 49 x^{4}$ |
| Frobenius angles: | $\pm0.401082902334$, $\pm0.537262832380$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-106 +2 \sqrt{5}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $3$ |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $55$ | $3905$ | $123805$ | $5486525$ | $280522000$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $7$ | $75$ | $361$ | $2283$ | $16692$ | $117975$ | $823711$ | $5765763$ | $40360357$ | $282454750$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 3 curves (of which all are hyperelliptic):
- $y^2=2 x^6+5 x^5+5 x^4+4 x^3+x^2+5 x+5$
- $y^2=2 x^6+4 x^4+3 x^2+6 x+6$
- $y^2=2 x^6+6 x^5+x^4+3 x^3+5 x^2+3 x$
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 \(\Q(\sqrt{-106 +2 \sqrt{5}})\). |
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_n | $2$ | 2.49.z_jt |