Basic invariants
| Dimension: | $16$ |
| Group: | $S_6$ |
| Conductor: | \(891\!\cdots\!000\)\(\medspace = 2^{36} \cdot 3^{12} \cdot 5^{12} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.1036800.1 |
| 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 6.2.1036800.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} - x^{4} + 6x^{3} - 2x^{2} - 4x - 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 137 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 137 }$:
\( x^{2} + 131x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 118 a + 5 + \left(93 a + 77\right)\cdot 137 + \left(97 a + 36\right)\cdot 137^{2} + \left(50 a + 102\right)\cdot 137^{3} + \left(121 a + 19\right)\cdot 137^{4} +O(137^{5})\)
| $r_{ 2 }$ |
$=$ |
\( 19 a + 28 + \left(43 a + 111\right)\cdot 137 + \left(39 a + 117\right)\cdot 137^{2} + \left(86 a + 34\right)\cdot 137^{3} + \left(15 a + 12\right)\cdot 137^{4} +O(137^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 51 + 99\cdot 137 + 51\cdot 137^{2} + 46\cdot 137^{3} + 97\cdot 137^{4} +O(137^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 29 a + 88 + \left(21 a + 131\right)\cdot 137 + \left(51 a + 24\right)\cdot 137^{2} + \left(96 a + 68\right)\cdot 137^{3} + \left(91 a + 92\right)\cdot 137^{4} +O(137^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 116 + 35\cdot 137 + 6\cdot 137^{2} + 112\cdot 137^{3} + 53\cdot 137^{4} +O(137^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 108 a + 125 + \left(115 a + 92\right)\cdot 137 + \left(85 a + 36\right)\cdot 137^{2} + \left(40 a + 47\right)\cdot 137^{3} + \left(45 a + 135\right)\cdot 137^{4} +O(137^{5})\)
| |
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.