Invariants
| Base field: | $\F_{11}$ |
| Dimension: | $3$ |
| L-polynomial: | $1 - 2 x + 25 x^{2} - 48 x^{3} + 275 x^{4} - 242 x^{5} + 1331 x^{6}$ |
| Frobenius angles: | $\pm0.284450250490$, $\pm0.530320729139$, $\pm0.573861780645$ |
| Angle rank: | $3$ (numerical) |
| Number field: | 6.0.66478640.1 |
| Galois group: | $S_4\times C_2$ |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1340$ | $2578160$ | $2355198740$ | $3106208418560$ | $4211071351773500$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $10$ | $168$ | $1330$ | $14492$ | $162350$ | $1772664$ | $19464070$ | $214326972$ | $2358123730$ | $25937676248$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 8 hyperelliptic curves, but it is unknown how many Jacobians of non-hyperelliptic curves it contains:
- $y^2=10 x^8+2 x^7+10 x^6+9 x^5+5 x^4+2 x^3+7 x^2+10 x+10$
- $y^2=9 x^8+x^6+2 x^5+x^4+6 x^3+7 x^2+7 x+8$
- $y^2=5 x^8+10 x^7+4 x^6+2 x^5+10 x^4+4 x^3+8 x^2+6$
- $y^2=3 x^8+3 x^7+7 x^4+2 x^3+6 x^2+x+5$
- $y^2=6 x^8+x^7+7 x^6+5 x^5+10 x^4+9 x^3+x+8$
- $y^2=7 x^8+2 x^7+10 x^5+5 x^4+7 x^3+9 x^2+3 x+9$
- $y^2=10 x^8+2 x^7+9 x^6+2 x^5+3 x^4+7 x^3+10 x^2+8 x+8$
- $y^2=6 x^8+4 x^7+10 x^5+2 x^3+2 x^2+4 x+6$
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 6.0.66478640.1. |
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.c_z_bw | $2$ | (not in LMFDB) |