Invariants
| Base field: | $\F_{7}$ |
| Dimension: | $3$ |
| L-polynomial: | $1 + 2 x + 9 x^{2} + 38 x^{3} + 63 x^{4} + 98 x^{5} + 343 x^{6}$ |
| Frobenius angles: | $\pm0.402719837316$, $\pm0.421780272365$, $\pm0.873538484449$ |
| Angle rank: | $3$ (numerical) |
| Number field: | 6.0.61594048.1 |
| Galois group: | $S_4\times C_2$ |
| 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: | $3$ |
| Slopes: | $[0, 0, 0, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $554$ | $154012$ | $49272206$ | $13366393456$ | $4578586162394$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $10$ | $64$ | $412$ | $2316$ | $16200$ | $118036$ | $825170$ | $5775132$ | $40327054$ | $282438364$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobian of 1 hyperelliptic curve, but it is unknown how many Jacobians of non-hyperelliptic curves it contains:
- $y^2=x^8+x^5+3 x^4+2 x^3+4 x^2+6 x+3$
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 6.0.61594048.1. |
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_j_abm | $2$ | (not in LMFDB) |