Invariants
| Base field: | $\F_{11}$ |
| Dimension: | $3$ |
| L-polynomial: | $1 - 10 x + 57 x^{2} - 216 x^{3} + 627 x^{4} - 1210 x^{5} + 1331 x^{6}$ |
| Frobenius angles: | $\pm0.186627373573$, $\pm0.254013931852$, $\pm0.507502273990$ |
| Angle rank: | $3$ (numerical) |
| Number field: | 6.0.18289840.1 |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $580$ | $2002160$ | $2473775980$ | $3174656930560$ | $4213985325794500$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $2$ | $136$ | $1394$ | $14812$ | $162462$ | $1776472$ | $19483998$ | $214301628$ | $2357855738$ | $25937484216$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 7 hyperelliptic curves, but it is unknown how many Jacobians of non-hyperelliptic curves it contains:
- $y^2=6 x^8+2 x^7+10 x^6+8 x^5+5 x^4+x^3+10 x+10$
- $y^2=6 x^8+7 x^7+2 x^6+x^5+3 x^3+9 x^2+9 x+2$
- $y^2=2 x^8+10 x^7+4 x^6+9 x^5+9 x^4+8 x^3+7 x^2+6 x+10$
- $y^2=6 x^8+6 x^7+4 x^6+6 x^5+3 x^4+7 x^3+7 x^2+4 x+9$
- $y^2=8 x^8+6 x^6+8 x^5+5 x^4+4 x^2+9 x$
- $y^2=2 x^8+8 x^7+8 x^6+x^5+9 x^4+10 x^3+5 x^2+x+2$
- $y^2=10 x^8+x^7+10 x^6+x^5+5 x^4+2 x^3+7 x^2+8 x+8$
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 6.0.18289840.1. |
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.k_cf_ii | $2$ | (not in LMFDB) |