Basic invariants
| Dimension: | $5$ |
| Group: | $S_6$ |
| Conductor: | \(4723920\)\(\medspace = 2^{4} \cdot 3^{10} \cdot 5 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.4723920.2 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_6$ |
| Parity: | even |
| Determinant: | 1.5.2t1.a.a |
| Projective image: | $S_6$ |
| Projective stem field: | Galois closure of 6.2.4723920.2 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} + 3x^{4} - 4x^{3} + 9x^{2} - 6x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 149 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 149 }$:
\( x^{2} + 145x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 102 a + 4 + \left(126 a + 81\right)\cdot 149 + \left(113 a + 123\right)\cdot 149^{2} + \left(88 a + 112\right)\cdot 149^{3} + \left(125 a + 8\right)\cdot 149^{4} +O(149^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 47 a + 114 + \left(22 a + 38\right)\cdot 149 + \left(35 a + 5\right)\cdot 149^{2} + \left(60 a + 56\right)\cdot 149^{3} + \left(23 a + 124\right)\cdot 149^{4} +O(149^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 87 a + 64 + \left(103 a + 96\right)\cdot 149 + \left(57 a + 98\right)\cdot 149^{2} + \left(62 a + 136\right)\cdot 149^{3} + \left(125 a + 85\right)\cdot 149^{4} +O(149^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 62 a + 114 + \left(45 a + 125\right)\cdot 149 + \left(91 a + 76\right)\cdot 149^{2} + \left(86 a + 30\right)\cdot 149^{3} + \left(23 a + 78\right)\cdot 149^{4} +O(149^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 104 + 57\cdot 149 + 6\cdot 149^{2} + 90\cdot 149^{3} + 85\cdot 149^{4} +O(149^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 47 + 47\cdot 149 + 136\cdot 149^{2} + 20\cdot 149^{3} + 64\cdot 149^{4} +O(149^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $5$ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $-1$ |
| $15$ | $2$ | $(1,2)$ | $3$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $40$ | $3$ | $(1,2,3)$ | $2$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $90$ | $4$ | $(1,2,3,4)$ | $1$ |
| $144$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $-1$ |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.