Invariants
| Base field: | $\F_{3^{2}}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 6 x + 22 x^{2} - 54 x^{3} + 81 x^{4}$ |
| Frobenius angles: | $\pm0.162381404600$, $\pm0.459361842123$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.7600.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $6$ |
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: | $2$ |
| Slopes: | $[0, 0, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $44$ | $7216$ | $545996$ | $42603264$ | $3493517324$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $4$ | $90$ | $748$ | $6494$ | $59164$ | $533946$ | $4790356$ | $43047614$ | $387389332$ | $3486773850$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 6 curves (of which all are hyperelliptic):
- $y^2=(a+2) x^6+x^5+2 x^4+x^3+x^2+(2 a+1) x+1$
- $y^2=a x^6+(a+1) x^5+(2 a+2) x^4+(a+1) x^3+a x^2+x+2 a+1$
- $y^2=a x^6+a x^5+(a+2) x^4+(2 a+1) x^3+2 a x^2+(a+1) x$
- $y^2=(2 a+1) x^6+2 a x^5+a x^4+(2 a+1) x^3+(a+2) x^2+x+2 a$
- $y^2=(2 a+1) x^6+a x^5+a x^4+(2 a+1) x^3+2 a x^2+2 x+2$
- $y^2=(2 a+1) x^5+a x^4+a x^3+(2 a+1) x+2 a+1$
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 4.0.7600.1. |
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 |
|---|---|---|
| 2.9.g_w | $2$ | 2.81.i_ac |