Basic invariants
| Dimension: | $5$ |
| Group: | $S_6$ |
| Conductor: | \(2678787049\)\(\medspace = 73^{2} \cdot 709^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.51757.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 12T183 |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $S_6$ |
| Projective stem field: | Galois closure of 6.2.51757.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{3} - 2x^{2} + x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 113 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 113 }$:
\( x^{2} + 101x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 23 a + 80 + \left(43 a + 75\right)\cdot 113 + \left(104 a + 69\right)\cdot 113^{2} + \left(a + 98\right)\cdot 113^{3} + \left(29 a + 104\right)\cdot 113^{4} +O(113^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 74 a + 93 + \left(81 a + 50\right)\cdot 113 + \left(61 a + 28\right)\cdot 113^{2} + \left(101 a + 41\right)\cdot 113^{3} + \left(7 a + 22\right)\cdot 113^{4} +O(113^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 13 a + 71 + \left(5 a + 52\right)\cdot 113 + \left(54 a + 58\right)\cdot 113^{2} + \left(96 a + 13\right)\cdot 113^{3} + \left(96 a + 74\right)\cdot 113^{4} +O(113^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 39 a + 77 + \left(31 a + 52\right)\cdot 113 + \left(51 a + 9\right)\cdot 113^{2} + \left(11 a + 68\right)\cdot 113^{3} + \left(105 a + 15\right)\cdot 113^{4} +O(113^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 90 a + 17 + \left(69 a + 6\right)\cdot 113 + \left(8 a + 36\right)\cdot 113^{2} + \left(111 a + 17\right)\cdot 113^{3} + \left(83 a + 112\right)\cdot 113^{4} +O(113^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 100 a + 1 + \left(107 a + 101\right)\cdot 113 + \left(58 a + 23\right)\cdot 113^{2} + \left(16 a + 100\right)\cdot 113^{3} + \left(16 a + 9\right)\cdot 113^{4} +O(113^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $5$ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $-3$ |
| $15$ | $2$ | $(1,2)$ | $1$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $2$ |
| $40$ | $3$ | $(1,2,3)$ | $-1$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $90$ | $4$ | $(1,2,3,4)$ | $-1$ |
| $144$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.