Invariants
| Base field: | $\F_{7}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + x + 7 x^{3} + 49 x^{4}$ |
| Frobenius angles: | $\pm0.287578126919$, $\pm0.799386933267$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-54 +2 \sqrt{57}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $4$ |
| Cyclic group of points: | yes |
This isogeny class is simple and geometrically simple, primitive, not ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
| $p$-rank: | $1$ |
| Slopes: | $[0, 1/2, 1/2, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $58$ | $2436$ | $125512$ | $6177696$ | $278982958$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $9$ | $49$ | $366$ | $2569$ | $16599$ | $117754$ | $821193$ | $5760433$ | $40367490$ | $282472729$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 4 curves (of which all are hyperelliptic):
- $y^2=2 x^6+3 x^5+x^4+6 x^3+6 x^2+2 x+4$
- $y^2=5 x^6+5 x^5+6 x^4+5 x^3+4 x^2+6 x+2$
- $y^2=3 x^6+2 x^4+x^3+5 x^2+4 x+4$
- $y^2=5 x^6+2 x^5+6 x^4+3 x^3+2 x^2+4 x+1$
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 \(\Q(\sqrt{-54 +2 \sqrt{57}})\). |
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_a | $2$ | 2.49.ab_dg |