Invariants
| Base field: | $\F_{7}$ |
| Dimension: | $3$ |
| L-polynomial: | $1 + 2 x + 19 x^{2} + 26 x^{3} + 133 x^{4} + 98 x^{5} + 343 x^{6}$ |
| Frobenius angles: | $\pm0.429026937724$, $\pm0.541558213987$, $\pm0.655350132092$ |
| Angle rank: | $3$ (numerical) |
| Number field: | 6.0.350726848.1 |
| Galois group: | $S_4\times C_2$ |
| Isomorphism classes: | 18 |
| 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: | $3$ |
| Slopes: | $[0, 0, 0, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $622$ | $230140$ | $37182538$ | $13223844400$ | $4773565671982$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $10$ | $84$ | $316$ | $2292$ | $16900$ | $117624$ | $824414$ | $5766876$ | $40341754$ | $282469104$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 2 hyperelliptic curves, but it is unknown how many Jacobians of non-hyperelliptic curves it contains:
- $y^2=x^8+x^5+x^4+3 x^3+5 x+1$
- $y^2=x^8+x^6+2 x^5+x^4+3 x^3+2 x^2+5 x+3$
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 6.0.350726848.1. |
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.ac_t_aba | $2$ | (not in LMFDB) |