Invariants
| Base field: | $\F_{7}$ |
| Dimension: | $3$ |
| L-polynomial: | $1 - 4 x + 14 x^{2} - 30 x^{3} + 98 x^{4} - 196 x^{5} + 343 x^{6}$ |
| Frobenius angles: | $\pm0.209171554878$, $\pm0.350275697141$, $\pm0.662651760947$ |
| Angle rank: | $3$ (numerical) |
| Number field: | 6.0.259687832.1 |
| Galois group: | $S_4\times C_2$ |
| Isomorphism classes: | 48 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $226$ | $155036$ | $41876896$ | $14800356704$ | $4808804794546$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $4$ | $62$ | $358$ | $2562$ | $17024$ | $116792$ | $824240$ | $5769026$ | $40340410$ | $282455742$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 6 hyperelliptic curves, but it is unknown how many Jacobians of non-hyperelliptic curves it contains:
- $y^2=x^7+x^5+2 x^2+4 x+2$
- $y^2=x^7+x^6+6 x^3+x^2+3 x+1$
- $y^2=x^8+x^3+2 x^2+3 x+5$
- $y^2=x^8+2 x^5+x^3+3 x^2+6 x+3$
- $y^2=x^8+x^6+x^5+2 x^2+3 x+3$
- $y^2=x^8+x^6+2 x^5+2 x^4+4 x^3+6 x^2+4 x+6$
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 6.0.259687832.1. |
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_o_be | $2$ | (not in LMFDB) |