Basic invariants
| Dimension: | $10$ |
| Group: | $S_6$ |
| Conductor: | \(16764094652917824\)\(\medspace = 2^{6} \cdot 3^{12} \cdot 149^{4} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.193104.1 |
| 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 6.2.193104.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} + 2x^{4} + x^{3} - 2x^{2} + x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 139 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 139 }$:
\( x^{2} + 138x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 136 a + 125 + \left(110 a + 62\right)\cdot 139 + \left(135 a + 120\right)\cdot 139^{2} + \left(78 a + 121\right)\cdot 139^{3} + \left(43 a + 107\right)\cdot 139^{4} +O(139^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 3 a + 122 + \left(28 a + 37\right)\cdot 139 + \left(3 a + 6\right)\cdot 139^{2} + \left(60 a + 65\right)\cdot 139^{3} + \left(95 a + 72\right)\cdot 139^{4} +O(139^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 93 a + 72 + \left(12 a + 137\right)\cdot 139 + \left(7 a + 84\right)\cdot 139^{2} + \left(109 a + 52\right)\cdot 139^{3} + \left(55 a + 46\right)\cdot 139^{4} +O(139^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 22 a + 95 + \left(133 a + 74\right)\cdot 139 + \left(116 a + 1\right)\cdot 139^{2} + \left(2 a + 138\right)\cdot 139^{3} + \left(55 a + 2\right)\cdot 139^{4} +O(139^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 117 a + 117 + \left(5 a + 46\right)\cdot 139 + \left(22 a + 124\right)\cdot 139^{2} + \left(136 a + 23\right)\cdot 139^{3} + \left(83 a + 55\right)\cdot 139^{4} +O(139^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 46 a + 26 + \left(126 a + 57\right)\cdot 139 + \left(131 a + 79\right)\cdot 139^{2} + \left(29 a + 15\right)\cdot 139^{3} + \left(83 a + 132\right)\cdot 139^{4} +O(139^{5})\)
|
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.