Basic invariants
| Dimension: | $5$ |
| Group: | $S_6$ |
| Conductor: | \(336069\)\(\medspace = 3^{6} \cdot 461 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.336069.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_6$ |
| Parity: | even |
| Determinant: | 1.461.2t1.a.a |
| Projective image: | $S_6$ |
| Projective stem field: | Galois closure of 6.2.336069.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 3x^{5} + 3x^{4} + 2x^{3} - 6x^{2} + 3x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 127 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 127 }$:
\( x^{2} + 126x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 97 a + 77 + \left(40 a + 81\right)\cdot 127 + \left(86 a + 81\right)\cdot 127^{2} + \left(123 a + 68\right)\cdot 127^{3} + \left(4 a + 66\right)\cdot 127^{4} +O(127^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 114 a + 53 + \left(45 a + 117\right)\cdot 127 + \left(66 a + 125\right)\cdot 127^{2} + \left(5 a + 93\right)\cdot 127^{3} + \left(20 a + 16\right)\cdot 127^{4} +O(127^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 25 a + 71 + \left(27 a + 52\right)\cdot 127 + \left(120 a + 30\right)\cdot 127^{2} + \left(68 a + 65\right)\cdot 127^{3} + \left(74 a + 29\right)\cdot 127^{4} +O(127^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 13 a + 40 + \left(81 a + 49\right)\cdot 127 + \left(60 a + 19\right)\cdot 127^{2} + \left(121 a + 33\right)\cdot 127^{3} + \left(106 a + 31\right)\cdot 127^{4} +O(127^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 102 a + 96 + \left(99 a + 54\right)\cdot 127 + \left(6 a + 123\right)\cdot 127^{2} + \left(58 a + 13\right)\cdot 127^{3} + \left(52 a + 35\right)\cdot 127^{4} +O(127^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 30 a + 47 + \left(86 a + 25\right)\cdot 127 + 40 a\cdot 127^{2} + \left(3 a + 106\right)\cdot 127^{3} + \left(122 a + 74\right)\cdot 127^{4} +O(127^{5})\)
|
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.