Invariants
| Base field: | $\F_{7}$ |
| Dimension: | $3$ |
| L-polynomial: | $1 - 4 x + 16 x^{2} - 47 x^{3} + 112 x^{4} - 196 x^{5} + 343 x^{6}$ |
| Frobenius angles: | $\pm0.157280763488$, $\pm0.433397922104$, $\pm0.607782259013$ |
| Angle rank: | $3$ (numerical) |
| Number field: | 6.0.940332987.1 |
| Galois group: | $S_4\times C_2$ |
| Isomorphism classes: | 40 |
| Cyclic group of points: | no |
| Non-cyclic primes: | $3$ |
This isogeny class is simple and geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable.
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $225$ | $161775$ | $38946825$ | $13576966875$ | $4814211049875$ |
Point counts of the (virtual) curve
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $4$ | $66$ | $331$ | $2354$ | $17044$ | $118233$ | $825808$ | $5772962$ | $40346044$ | $282418266$ |
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 6.0.940332987.1. |
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.e_q_bv | $2$ | (not in LMFDB) |