Invariants
| Base field: | $\F_{7}$ |
| Dimension: | $3$ |
| L-polynomial: | $1 - 3 x + 18 x^{2} - 39 x^{3} + 126 x^{4} - 147 x^{5} + 343 x^{6}$ |
| Frobenius angles: | $\pm0.261695992286$, $\pm0.460186622800$, $\pm0.576642003019$ |
| Angle rank: | $3$ (numerical) |
| Number field: | 6.0.6252032304.1 |
| Galois group: | $S_4\times C_2$ |
| Isomorphism classes: | 21 |
| Cyclic group of points: | yes |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $299$ | $202423$ | $42605108$ | $13577927571$ | $4814963325029$ |
Point counts of the (virtual) curve
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $5$ | $77$ | $362$ | $2357$ | $17045$ | $118088$ | $821567$ | $5759285$ | $40347074$ | $282472937$ |
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 6.0.6252032304.1. |
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.d_s_bn | $2$ | (not in LMFDB) |