Invariants
| Base field: | $\F_{7}$ |
| Dimension: | $3$ |
| L-polynomial: | $1 + 3 x + 21 x^{2} + 41 x^{3} + 147 x^{4} + 147 x^{5} + 343 x^{6}$ |
| Frobenius angles: | $\pm0.467938051354$, $\pm0.539363585426$, $\pm0.683148769828$ |
| Angle rank: | $3$ (numerical) |
| Number field: | 6.0.98828343.1 |
| Galois group: | $A_4\times C_2$ |
| Cyclic group of points: | yes |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $703$ | $225663$ | $36077257$ | $13209635031$ | $4774393617913$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $11$ | $83$ | $305$ | $2291$ | $16901$ | $118067$ | $824744$ | $5759363$ | $40343651$ | $282533243$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobian of 1 hyperelliptic curve, but it is unknown how many Jacobians of non-hyperelliptic curves it contains:
- $y^2=x^7+3 x^5+5 x^2+2 x+2$
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 6.0.98828343.1. |
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.ad_v_abp | $2$ | (not in LMFDB) |