Basic invariants
| Dimension: | $3$ |
| Group: | $S_4\times C_2$ |
| Conductor: | \(21824\)\(\medspace = 2^{6} \cdot 11 \cdot 31 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.676544.2 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_4\times C_2$ |
| Parity: | odd |
| Determinant: | 1.1364.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.2.3751.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 6x^{4} + 13x^{2} - 11 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 131 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 131 }$:
\( x^{2} + 127x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 8 a + 115 + \left(26 a + 82\right)\cdot 131 + \left(126 a + 22\right)\cdot 131^{2} + \left(11 a + 39\right)\cdot 131^{3} + \left(128 a + 77\right)\cdot 131^{4} + \left(81 a + 96\right)\cdot 131^{5} + \left(69 a + 32\right)\cdot 131^{6} +O(131^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 53 + 14\cdot 131 + 29\cdot 131^{2} + 11\cdot 131^{3} + 35\cdot 131^{4} + 72\cdot 131^{5} + 86\cdot 131^{6} +O(131^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 98 a + 66 + \left(48 a + 82\right)\cdot 131 + \left(107 a + 71\right)\cdot 131^{2} + \left(19 a + 79\right)\cdot 131^{3} + \left(123 a + 25\right)\cdot 131^{4} + \left(119 a + 18\right)\cdot 131^{5} + \left(72 a + 45\right)\cdot 131^{6} +O(131^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 123 a + 16 + \left(104 a + 48\right)\cdot 131 + \left(4 a + 108\right)\cdot 131^{2} + \left(119 a + 91\right)\cdot 131^{3} + \left(2 a + 53\right)\cdot 131^{4} + \left(49 a + 34\right)\cdot 131^{5} + \left(61 a + 98\right)\cdot 131^{6} +O(131^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( 78 + 116\cdot 131 + 101\cdot 131^{2} + 119\cdot 131^{3} + 95\cdot 131^{4} + 58\cdot 131^{5} + 44\cdot 131^{6} +O(131^{7})\)
|
| $r_{ 6 }$ | $=$ |
\( 33 a + 65 + \left(82 a + 48\right)\cdot 131 + \left(23 a + 59\right)\cdot 131^{2} + \left(111 a + 51\right)\cdot 131^{3} + \left(7 a + 105\right)\cdot 131^{4} + \left(11 a + 112\right)\cdot 131^{5} + \left(58 a + 85\right)\cdot 131^{6} +O(131^{7})\)
|
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 |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-3$ |
| $3$ | $2$ | $(3,6)$ | $1$ |
| $3$ | $2$ | $(2,5)(3,6)$ | $-1$ |
| $6$ | $2$ | $(1,2)(4,5)$ | $1$ |
| $6$ | $2$ | $(1,2)(3,6)(4,5)$ | $-1$ |
| $8$ | $3$ | $(1,3,2)(4,6,5)$ | $0$ |
| $6$ | $4$ | $(2,3,5,6)$ | $1$ |
| $6$ | $4$ | $(1,4)(2,3,5,6)$ | $-1$ |
| $8$ | $6$ | $(1,3,5,4,6,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.