Invariants
| Base field: | $\F_{7}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + x - 8 x^{2} + 7 x^{3} + 49 x^{4}$ |
| Frobenius angles: | $\pm0.206452615016$, $\pm0.946518960900$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-22 +2 \sqrt{89}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $1$ |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $50$ | $1700$ | $134600$ | $5780000$ | $289797750$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $9$ | $33$ | $390$ | $2409$ | $17239$ | $117786$ | $824553$ | $5761521$ | $40342890$ | $282445993$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobian of 1 curve (which is hyperelliptic):
- $y^2=4 x^6+3 x^5+x^4+5 x^3+6 x^2+5$
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 \(\Q(\sqrt{-22 +2 \sqrt{89}})\). |
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.ab_ai | $2$ | 2.49.ar_fs |