Invariants
| Base field: | $\F_{7}$ |
| Dimension: | $3$ |
| L-polynomial: | $1 - 4 x + 22 x^{2} - 53 x^{3} + 154 x^{4} - 196 x^{5} + 343 x^{6}$ |
| Frobenius angles: | $\pm0.273100582737$, $\pm0.424762016427$, $\pm0.542209666657$ |
| Angle rank: | $3$ (numerical) |
| Number field: | 6.0.1202179347.1 |
| Galois group: | $S_4\times C_2$ |
| Isomorphism classes: | 10 |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $267$ | $206391$ | $45534447$ | $13561746219$ | $4754963043717$ |
Point counts of the (virtual) curve
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $4$ | $78$ | $385$ | $2354$ | $16834$ | $117945$ | $822154$ | $5759954$ | $40352956$ | $282481758$ |
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 6.0.1202179347.1. |
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_w_cb | $2$ | (not in LMFDB) |