Basic invariants
| Dimension: | $2$ |
| Group: | $S_3$ |
| Conductor: | \(1436\)\(\medspace = 2^{2} \cdot 359 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 3.3.1436.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_3$ |
| Parity: | even |
| Determinant: | 1.1436.2t1.a.a |
| Projective image: | $S_3$ |
| Projective stem field: | Galois closure of 3.3.1436.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{3} - 11x - 12 \)
|
The roots of $f$ are computed in $\Q_{ 41 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 15 + 21\cdot 41 + 34\cdot 41^{2} + 7\cdot 41^{3} + 22\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 31 + 10\cdot 41 + 34\cdot 41^{2} + 23\cdot 41^{3} + 12\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 36 + 8\cdot 41 + 13\cdot 41^{2} + 9\cdot 41^{3} + 6\cdot 41^{4} +O(41^{5})\)
|
Generators of the action on the roots $ r_{ 1 }, r_{ 2 }, r_{ 3 } $
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $ r_{ 1 }, r_{ 2 }, r_{ 3 } $ | Character value |
| $1$ | $1$ | $()$ | $2$ |
| $3$ | $2$ | $(1,2)$ | $0$ |
| $2$ | $3$ | $(1,2,3)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.