Invariants
| Base field: | $\F_{7}$ |
| Dimension: | $3$ |
| L-polynomial: | $1 - 8 x + 37 x^{2} - 116 x^{3} + 259 x^{4} - 392 x^{5} + 343 x^{6}$ |
| Frobenius angles: | $\pm0.122700487902$, $\pm0.322080743167$, $\pm0.482502673746$ |
| Angle rank: | $3$ (numerical) |
| Number field: | 6.0.12178624.2 |
| Galois group: | $S_4\times C_2$ |
| Isomorphism classes: | 18 |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $124$ | $143344$ | $43874548$ | $13625707264$ | $4729062060364$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $0$ | $60$ | $372$ | $2364$ | $16740$ | $118092$ | $824600$ | $5766396$ | $40371672$ | $282555180$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 2 hyperelliptic curves, but it is unknown how many Jacobians of non-hyperelliptic curves it contains:
- $y^2=3 x^8+3 x^4+3 x^3+4 x^2+4 x+3$
- $y^2=3 x^8+x^6+x^5+x^4+x^3+3 x^2+5 x+5$
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 6.0.12178624.2. |
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.i_bl_em | $2$ | (not in LMFDB) |