Invariants
| Base field: | $\F_{3^{2}}$ |
| Dimension: | $3$ |
| L-polynomial: | $1 - x - 3 x^{2} + 45 x^{3} - 27 x^{4} - 81 x^{5} + 729 x^{6}$ |
| Frobenius angles: | $\pm0.225992293488$, $\pm0.372040413785$, $\pm0.941167390149$ |
| Angle rank: | $3$ (numerical) |
| Number field: | 6.0.112056007.1 |
| Galois group: | $S_4\times C_2$ |
| Cyclic group of points: | yes |
This isogeny class is simple and geometrically simple, primitive, not ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
| $p$-rank: | $1$ |
| Slopes: | $[0, 1/2, 1/2, 1/2, 1/2, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $663$ | $488631$ | $459401319$ | $284203914423$ | $206932639697823$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $9$ | $75$ | $855$ | $6603$ | $59349$ | $530343$ | $4784460$ | $43047795$ | $387395757$ | $3486660495$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 4 hyperelliptic curves, but it is unknown how many Jacobians of non-hyperelliptic curves it contains:
- $y^2=x^7+x^4+a x+1$
- $y^2=x^7+x^4+(2 a+1) x+1$
- $y^2=x^7+a x^5+(a+1) x^4+(a+1) x^3+2 x^2+2 x+2 a$
- $y^2=x^7+(2 a+1) x^5+(a+1) x^4+(2 a+2) x^3+x^2+2 x+2 a+1$
where $a$ is a root of the Conway polynomial.
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{3^{2}}$.
Endomorphism algebra over $\F_{3^{2}}$| The endomorphism algebra of this simple isogeny class is 6.0.112056007.1. |
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 |
|---|---|---|
| 3.9.b_ad_abt | $2$ | (not in LMFDB) |