Invariants
| Base field: | $\F_{7}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - x - 5 x^{2} - 7 x^{3} + 49 x^{4}$ |
| Frobenius angles: | $\pm0.125193012804$, $\pm0.762660685111$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-34 +2 \sqrt{77}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $2$ |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $37$ | $1961$ | $105931$ | $6002621$ | $280537552$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $7$ | $39$ | $307$ | $2499$ | $16692$ | $118299$ | $826105$ | $5765043$ | $40376881$ | $282485854$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 2 curves (of which all are hyperelliptic):
- $y^2=x^6+3 x^5+5 x^3+4 x^2+4 x+6$
- $y^2=x^6+4 x^4+6 x^3+6 x^2+2 x+2$
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 \(\Q(\sqrt{-34 +2 \sqrt{77}})\). |
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_af | $2$ | 2.49.al_ef |