Invariants
| Base field: | $\F_{7}$ |
| Dimension: | $3$ |
| L-polynomial: | $1 - 2 x + x^{2} + 22 x^{3} + 7 x^{4} - 98 x^{5} + 343 x^{6}$ |
| Frobenius angles: | $\pm0.244912085218$, $\pm0.319914035316$, $\pm0.840067488833$ |
| Angle rank: | $3$ (numerical) |
| Number field: | 6.0.38463424.1 |
| Galois group: | $S_4\times C_2$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $274$ | $117820$ | $48647878$ | $15055039600$ | $4721742428674$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $6$ | $48$ | $408$ | $2604$ | $16716$ | $117732$ | $819398$ | $5763996$ | $40352826$ | $282486828$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 3 hyperelliptic curves, but it is unknown how many Jacobians of non-hyperelliptic curves it contains:
- $y^2=x^8+x^6+x^5+3 x^4+5 x^3+4 x^2+2 x+5$
- $y^2=x^8+x^6+3 x^5+5 x^4+6 x^3+3 x^2+4 x+2$
- $y^2=x^8+3 x^6+x^5+x^4+3 x^3+6 x^2+x+3$
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 6.0.38463424.1. |
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.c_b_aw | $2$ | (not in LMFDB) |