Invariants
| Base field: | $\F_{7}$ |
| Dimension: | $3$ |
| L-polynomial: | $1 + 14 x^{2} - 7 x^{3} + 98 x^{4} + 343 x^{6}$ |
| Frobenius angles: | $\pm0.304538351939$, $\pm0.582546052305$, $\pm0.603602957694$ |
| Angle rank: | $3$ (numerical) |
| Number field: | 6.0.602651.1 |
| Galois group: | $A_4\times C_2$ |
| Cyclic group of points: | yes |
This isogeny class is simple and geometrically simple, primitive, not ordinary, and not supersingular. It is principally polarizable.
Newton polygon
| $p$-rank: | $0$ |
| Slopes: | $[1/3, 1/3, 1/3, 2/3, 2/3, 2/3]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $449$ | $207887$ | $37829597$ | $13847976731$ | $4887765324359$ |
Point counts of the (virtual) curve
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $8$ | $78$ | $323$ | $2402$ | $17298$ | $116817$ | $818742$ | $5770290$ | $40374188$ | $282446438$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains no Jacobian of a hyperelliptic curve, but it is unknown whether it contains a Jacobian of a non-hyperelliptic curve.
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 6.0.602651.1. |
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.a_o_h | $2$ | (not in LMFDB) |