Basic invariants
| Dimension: | $10$ |
| Group: | $S_6$ |
| Conductor: | \(60112432135143424\)\(\medspace = 2^{26} \cdot 173^{4} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.354304.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 30T164 |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $S_6$ |
| Projective stem field: | Galois closure of 6.2.354304.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 4x^{3} - 4x^{2} - 4x - 2 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 151 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 151 }$:
\( x^{2} + 149x + 6 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 145 a + 77 + \left(116 a + 47\right)\cdot 151 + \left(136 a + 145\right)\cdot 151^{2} + \left(81 a + 87\right)\cdot 151^{3} + \left(47 a + 144\right)\cdot 151^{4} +O(151^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 67 a + 122 + \left(132 a + 94\right)\cdot 151 + \left(131 a + 74\right)\cdot 151^{2} + \left(106 a + 19\right)\cdot 151^{3} + \left(11 a + 20\right)\cdot 151^{4} +O(151^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 84 a + 105 + \left(18 a + 141\right)\cdot 151 + \left(19 a + 54\right)\cdot 151^{2} + \left(44 a + 101\right)\cdot 151^{3} + \left(139 a + 87\right)\cdot 151^{4} +O(151^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 20 + 114\cdot 151 + 14\cdot 151^{2} + 40\cdot 151^{3} + 14\cdot 151^{4} +O(151^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 64 + 69\cdot 151 + 12\cdot 151^{2} + 89\cdot 151^{3} + 28\cdot 151^{4} +O(151^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 6 a + 65 + \left(34 a + 136\right)\cdot 151 + \left(14 a + 150\right)\cdot 151^{2} + \left(69 a + 114\right)\cdot 151^{3} + \left(103 a + 6\right)\cdot 151^{4} +O(151^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $10$ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $-2$ |
| $15$ | $2$ | $(1,2)$ | $2$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $-2$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $1$ |
| $40$ | $3$ | $(1,2,3)$ | $1$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $90$ | $4$ | $(1,2,3,4)$ | $0$ |
| $144$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $1$ |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.