Basic invariants
| Dimension: | $16$ |
| Group: | $S_6$ |
| Conductor: | \(641\!\cdots\!936\)\(\medspace = 2^{24} \cdot 3^{8} \cdot 17^{12} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.4.16036032.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 36T1252 |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $S_6$ |
| Projective stem field: | Galois closure of 6.4.16036032.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{4} - 10x^{3} - 11x^{2} + 6x + 5 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 125 a + 102 + \left(59 a + 59\right)\cdot 127 + \left(93 a + 94\right)\cdot 127^{2} + 61\cdot 127^{3} + \left(81 a + 36\right)\cdot 127^{4} +O(127^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 3 a + 104 + \left(15 a + 19\right)\cdot 127 + \left(58 a + 55\right)\cdot 127^{2} + \left(7 a + 9\right)\cdot 127^{3} + \left(32 a + 62\right)\cdot 127^{4} +O(127^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 56 + 86\cdot 127 + 65\cdot 127^{2} + 14\cdot 127^{3} + 15\cdot 127^{4} +O(127^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 2 a + 100 + \left(67 a + 121\right)\cdot 127 + 33 a\cdot 127^{2} + \left(126 a + 96\right)\cdot 127^{3} + \left(45 a + 116\right)\cdot 127^{4} +O(127^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 39 + 61\cdot 127 + 66\cdot 127^{2} + 113\cdot 127^{3} + 63\cdot 127^{4} +O(127^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 124 a + 107 + \left(111 a + 31\right)\cdot 127 + \left(68 a + 98\right)\cdot 127^{2} + \left(119 a + 85\right)\cdot 127^{3} + \left(94 a + 86\right)\cdot 127^{4} +O(127^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $16$ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $15$ | $2$ | $(1,2)$ | $0$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $-2$ |
| $40$ | $3$ | $(1,2,3)$ | $-2$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $90$ | $4$ | $(1,2,3,4)$ | $0$ |
| $144$ | $5$ | $(1,2,3,4,5)$ | $1$ |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.