Invariants
| Base field: | $\F_{7}$ |
| Dimension: | $3$ |
| L-polynomial: | $1 - x + 5 x^{2} - 30 x^{3} + 35 x^{4} - 49 x^{5} + 343 x^{6}$ |
| Frobenius angles: | $\pm0.120576034706$, $\pm0.572606806314$, $\pm0.671836283828$ |
| Angle rank: | $3$ (numerical) |
| Number field: | 6.0.13081136.1 |
| Galois group: | $S_4\times C_2$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2$ |
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: | $3$ |
| Slopes: | $[0, 0, 0, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $304$ | $141056$ | $32322496$ | $13787377664$ | $4870526907664$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $7$ | $59$ | $268$ | $2391$ | $17237$ | $117548$ | $824411$ | $5772623$ | $40364548$ | $282501459$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 8 hyperelliptic curves, but it is unknown how many Jacobians of non-hyperelliptic curves it contains:
- $y^2=x^7+x^5+6 x^3+x^2+4 x+4$
- $y^2=x^7+x^5+6 x^3+5 x^2+2 x+3$
- $y^2=x^7+x^5+6 x^3+6 x^2+4 x+2$
- $y^2=x^7+x^6+5 x^3+5 x^2+6 x+5$
- $y^2=x^7+x^6+3 x^4+x^3+x^2+5$
- $y^2=x^7+x^6+4 x^4+x^2+6 x+1$
- $y^2=x^7+x^6+4 x^4+3 x^3+5 x+2$
- $y^2=x^7+x^6+4 x^4+4 x^3+6 x+4$
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 6.0.13081136.1. |
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 |
|---|---|---|
| 3.7.b_f_be | $2$ | (not in LMFDB) |