Basic invariants
| Dimension: | $3$ |
| Group: | $S_4\times C_2$ |
| Conductor: | \(1219\)\(\medspace = 23 \cdot 53 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.28037.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_4\times C_2$ |
| Parity: | odd |
| Determinant: | 1.1219.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.2.64607.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} + 3x^{3} - 2x - 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$:
\( x^{2} + 16x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 14 a + 5 + \left(15 a + 9\right)\cdot 17 + \left(14 a + 15\right)\cdot 17^{2} + \left(a + 8\right)\cdot 17^{3} + \left(5 a + 5\right)\cdot 17^{4} + \left(11 a + 13\right)\cdot 17^{5} + \left(5 a + 8\right)\cdot 17^{6} + \left(a + 7\right)\cdot 17^{7} + \left(2 a + 9\right)\cdot 17^{8} +O(17^{9})\)
|
| $r_{ 2 }$ | $=$ |
\( 16 a + 5 + \left(7 a + 11\right)\cdot 17 + \left(2 a + 7\right)\cdot 17^{2} + \left(9 a + 7\right)\cdot 17^{3} + \left(11 a + 16\right)\cdot 17^{4} + \left(9 a + 11\right)\cdot 17^{5} + \left(8 a + 13\right)\cdot 17^{6} + \left(15 a + 15\right)\cdot 17^{7} + \left(2 a + 5\right)\cdot 17^{8} +O(17^{9})\)
|
| $r_{ 3 }$ | $=$ |
\( 14 + 12\cdot 17 + 8\cdot 17^{2} + 9\cdot 17^{4} + 13\cdot 17^{5} + 10\cdot 17^{6} + 16\cdot 17^{7} + 2\cdot 17^{8} +O(17^{9})\)
|
| $r_{ 4 }$ | $=$ |
\( a + 4 + \left(9 a + 3\right)\cdot 17 + \left(14 a + 2\right)\cdot 17^{2} + \left(7 a + 14\right)\cdot 17^{3} + \left(5 a + 1\right)\cdot 17^{4} + \left(7 a + 10\right)\cdot 17^{5} + \left(8 a + 12\right)\cdot 17^{6} + \left(a + 5\right)\cdot 17^{7} + \left(14 a + 10\right)\cdot 17^{8} +O(17^{9})\)
|
| $r_{ 5 }$ | $=$ |
\( 3 a + 2 + \left(a + 11\right)\cdot 17 + \left(2 a + 14\right)\cdot 17^{2} + \left(15 a + 12\right)\cdot 17^{3} + \left(11 a + 8\right)\cdot 17^{4} + \left(5 a + 2\right)\cdot 17^{5} + \left(11 a + 3\right)\cdot 17^{6} + \left(15 a + 3\right)\cdot 17^{7} + \left(14 a + 10\right)\cdot 17^{8} +O(17^{9})\)
|
| $r_{ 6 }$ | $=$ |
\( 6 + 3\cdot 17 + 2\cdot 17^{2} + 7\cdot 17^{3} + 9\cdot 17^{4} + 16\cdot 17^{5} + 17^{6} + 2\cdot 17^{7} + 12\cdot 17^{8} +O(17^{9})\)
|
Generators of the action on the roots $r_1, \ldots, r_{ 6 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 6 }$ | Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,2)(3,6)(4,5)$ | $-3$ |
| $3$ | $2$ | $(1,2)(3,6)$ | $-1$ |
| $3$ | $2$ | $(3,6)$ | $1$ |
| $6$ | $2$ | $(1,3)(2,6)$ | $1$ |
| $6$ | $2$ | $(1,2)(3,4)(5,6)$ | $-1$ |
| $8$ | $3$ | $(1,4,3)(2,5,6)$ | $0$ |
| $6$ | $4$ | $(1,6,2,3)$ | $1$ |
| $6$ | $4$ | $(1,2)(3,5,6,4)$ | $-1$ |
| $8$ | $6$ | $(1,4,3,2,5,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.