Invariants
| Base field: | $\F_{7}$ |
| Dimension: | $3$ |
| L-polynomial: | $1 + x + 15 x^{2} + 16 x^{3} + 105 x^{4} + 49 x^{5} + 343 x^{6}$ |
| Frobenius angles: | $\pm0.391971823475$, $\pm0.478126810968$, $\pm0.701088299494$ |
| Angle rank: | $3$ (numerical) |
| Number field: | 6.0.4081838848.1 |
| Galois group: | $S_4\times C_2$ |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $530$ | $210940$ | $40782440$ | $13639380400$ | $4650216584150$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $9$ | $79$ | $348$ | $2367$ | $16459$ | $117424$ | $827381$ | $5763471$ | $40337988$ | $282502279$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 5 hyperelliptic curves, but it is unknown how many Jacobians of non-hyperelliptic curves it contains:
- $y^2=x^7+x^5+2 x^3+4 x^2+2 x+5$
- $y^2=x^7+3 x^6+x^4+6 x^3+4 x^2+4 x+6$
- $y^2=x^7+3 x^6+4 x^4+5 x^3+3 x+3$
- $y^2=x^7+3 x^6+4 x^4+5 x^3+2 x^2+4 x+4$
- $y^2=x^7+3 x^6+6 x^4+4 x^2+5 x+1$
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 6.0.4081838848.1. |
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.ab_p_aq | $2$ | (not in LMFDB) |