Invariants
| Base field: | $\F_{7}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - x - 3 x^{2} - 7 x^{3} + 49 x^{4}$ |
| Frobenius angles: | $\pm0.157948536279$, $\pm0.742570314434$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-42 +2 \sqrt{69}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $4$ |
| Isomorphism classes: | 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $39$ | $2145$ | $107757$ | $6102525$ | $283215504$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $7$ | $43$ | $313$ | $2539$ | $16852$ | $118231$ | $826735$ | $5763331$ | $40363621$ | $282474718$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 4 curves (of which all are hyperelliptic):
- $y^2=4 x^6+2 x^5+5 x^4+4 x^3+x^2+4 x+5$
- $y^2=4 x^6+6 x^5+4 x^4+4 x^2+1$
- $y^2=2 x^6+3 x^4+5 x^3+2 x^2+4 x+2$
- $y^2=4 x^6+5 x^4+6 x^3+6 x^2+x+5$
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 \(\Q(\sqrt{-42 +2 \sqrt{69}})\). |
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_ad | $2$ | 2.49.ah_dp |