Invariants
| Base field: | $\F_{11}$ |
| Dimension: | $3$ |
| L-polynomial: | $1 + 6 x + 36 x^{2} + 126 x^{3} + 396 x^{4} + 726 x^{5} + 1331 x^{6}$ |
| Frobenius angles: | $\pm0.464098343535$, $\pm0.574831949550$, $\pm0.786901481039$ |
| Angle rank: | $3$ (numerical) |
| Number field: | 6.0.11812955112.1 |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $2622$ | $2375532$ | $2266918272$ | $3107518928352$ | $4164941119657182$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $18$ | $158$ | $1278$ | $14498$ | $160578$ | $1775768$ | $19484406$ | $214332098$ | $2358042138$ | $25937040878$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 11 hyperelliptic curves, but it is unknown how many Jacobians of non-hyperelliptic curves it contains:
- $y^2=9 x^8+7 x^7+6 x^6+8 x^5+8 x^4+x^3+x^2+6 x+3$
- $y^2=5 x^8+2 x^7+10 x^6+10 x^5+8 x^4+8 x^3+6 x^2+9$
- $y^2=5 x^8+8 x^7+2 x^6+10 x^5+4 x^4+8 x^3+7 x^2+8 x+6$
- $y^2=4 x^7+x^6+2 x^5+x^4+10 x^3+5 x^2+3 x+5$
- $y^2=4 x^8+7 x^7+8 x^6+6 x^5+x^4+2 x^3+x^2+7 x+3$
- $y^2=4 x^8+10 x^7+2 x^6+3 x^5+10 x^4+7 x^3+x^2+4 x+3$
- $y^2=2 x^8+5 x^7+8 x^6+x^5+8 x^4+2 x^3+5 x^2+3$
- $y^2=3 x^8+9 x^7+6 x^5+6 x^4+6 x^3+2 x^2+x+4$
- $y^2=3 x^8+4 x^7+10 x^6+5 x^5+10 x^4+4 x^3+8 x^2+5$
- $y^2=7 x^8+7 x^6+8 x^5+7 x^4+7 x^3+9 x^2+4 x+3$
- $y^2=9 x^8+2 x^7+5 x^2+6 x+5$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{11}$.
Endomorphism algebra over $\F_{11}$| The endomorphism algebra of this simple isogeny class is 6.0.11812955112.1. |
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 |
|---|---|---|
| 3.11.ag_bk_aew | $2$ | (not in LMFDB) |