Basic invariants
| Dimension: | $3$ |
| Group: | $S_4\times C_2$ |
| Conductor: | \(9724\)\(\medspace = 2^{2} \cdot 11 \cdot 13 \cdot 17 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.427856.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_4\times C_2$ |
| Parity: | odd |
| Determinant: | 1.2431.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.2.2149004.2 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} - x^{3} - 2x^{2} - 7x - 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 7 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 7 }$:
\( x^{2} + 6x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 6 a + 4 + \left(4 a + 2\right)\cdot 7 + \left(4 a + 2\right)\cdot 7^{2} + \left(a + 6\right)\cdot 7^{3} + 6 a\cdot 7^{4} + \left(6 a + 3\right)\cdot 7^{5} + \left(2 a + 5\right)\cdot 7^{6} + 4\cdot 7^{7} + \left(a + 1\right)\cdot 7^{8} + \left(3 a + 2\right)\cdot 7^{9} +O(7^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 6 + 4\cdot 7 + 5\cdot 7^{2} + 7^{3} + 4\cdot 7^{4} + 7^{5} + 7^{6} + 2\cdot 7^{7} + 5\cdot 7^{8} + 5\cdot 7^{9} +O(7^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 6 a + 3 + 5 a\cdot 7 + 2\cdot 7^{2} + 5 a\cdot 7^{3} + \left(4 a + 1\right)\cdot 7^{4} + \left(a + 5\right)\cdot 7^{5} + \left(4 a + 6\right)\cdot 7^{6} + \left(5 a + 6\right)\cdot 7^{7} + \left(4 a + 2\right)\cdot 7^{8} + \left(4 a + 5\right)\cdot 7^{9} +O(7^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 4 + 4\cdot 7 + 4\cdot 7^{2} + 4\cdot 7^{3} + 7^{4} + 5\cdot 7^{5} + 3\cdot 7^{6} + 3\cdot 7^{7} + 6\cdot 7^{8} + 4\cdot 7^{9} +O(7^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( a + 3 + \left(2 a + 1\right)\cdot 7 + \left(2 a + 2\right)\cdot 7^{2} + \left(5 a + 3\right)\cdot 7^{3} + 5\cdot 7^{4} + 3\cdot 7^{5} + \left(4 a + 1\right)\cdot 7^{6} + \left(6 a + 2\right)\cdot 7^{7} + \left(5 a + 2\right)\cdot 7^{8} + \left(3 a + 4\right)\cdot 7^{9} +O(7^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( a + 2 + a\cdot 7 + \left(6 a + 4\right)\cdot 7^{2} + \left(a + 4\right)\cdot 7^{3} + 2 a\cdot 7^{4} + \left(5 a + 2\right)\cdot 7^{5} + \left(2 a + 2\right)\cdot 7^{6} + \left(a + 1\right)\cdot 7^{7} + \left(2 a + 2\right)\cdot 7^{8} + \left(2 a + 5\right)\cdot 7^{9} +O(7^{10})\)
|
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 |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,3)(2,4)(5,6)$ | $-3$ |
| $3$ | $2$ | $(1,3)(2,4)$ | $-1$ |
| $3$ | $2$ | $(2,4)$ | $1$ |
| $6$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $6$ | $2$ | $(1,3)(2,5)(4,6)$ | $-1$ |
| $8$ | $3$ | $(1,5,2)(3,6,4)$ | $0$ |
| $6$ | $4$ | $(1,4,3,2)$ | $1$ |
| $6$ | $4$ | $(1,3)(2,6,4,5)$ | $-1$ |
| $8$ | $6$ | $(1,5,2,3,6,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.