Basic invariants
Dimension: | $10$ |
Group: | $S_6$ |
Conductor: | \(54419558400000000\)\(\medspace = 2^{18} \cdot 3^{12} \cdot 5^{8} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 6.2.648000.1 |
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 6.2.648000.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} + 2x^{4} - 2x^{3} - 3x^{2} - 6x - 3 \) . |
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:
$r_{ 1 }$ | $=$ |
\( 41 a + 70 + \left(43 a + 31\right)\cdot 137 + \left(73 a + 12\right)\cdot 137^{2} + \left(136 a + 9\right)\cdot 137^{3} + \left(94 a + 98\right)\cdot 137^{4} +O(137^{5})\)
$r_{ 2 }$ |
$=$ |
\( 116 a + 57 + \left(90 a + 11\right)\cdot 137 + \left(14 a + 57\right)\cdot 137^{2} + \left(99 a + 99\right)\cdot 137^{3} + \left(66 a + 19\right)\cdot 137^{4} +O(137^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 102 a + 55 + \left(101 a + 132\right)\cdot 137 + \left(21 a + 61\right)\cdot 137^{2} + \left(60 a + 86\right)\cdot 137^{3} + \left(44 a + 96\right)\cdot 137^{4} +O(137^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 35 a + 119 + \left(35 a + 92\right)\cdot 137 + \left(115 a + 90\right)\cdot 137^{2} + \left(76 a + 14\right)\cdot 137^{3} + \left(92 a + 29\right)\cdot 137^{4} +O(137^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 96 a + 42 + \left(93 a + 113\right)\cdot 137 + \left(63 a + 134\right)\cdot 137^{2} + 69\cdot 137^{3} + \left(42 a + 120\right)\cdot 137^{4} +O(137^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 21 a + 68 + \left(46 a + 29\right)\cdot 137 + \left(122 a + 54\right)\cdot 137^{2} + \left(37 a + 131\right)\cdot 137^{3} + \left(70 a + 46\right)\cdot 137^{4} +O(137^{5})\)
| |
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 |
$1$ | $1$ | $()$ | $10$ |
$15$ | $2$ | $(1,2)(3,4)(5,6)$ | $2$ |
$15$ | $2$ | $(1,2)$ | $-2$ |
$45$ | $2$ | $(1,2)(3,4)$ | $-2$ |
$40$ | $3$ | $(1,2,3)(4,5,6)$ | $1$ |
$40$ | $3$ | $(1,2,3)$ | $1$ |
$90$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
$90$ | $4$ | $(1,2,3,4)$ | $0$ |
$144$ | $5$ | $(1,2,3,4,5)$ | $0$ |
$120$ | $6$ | $(1,2,3,4,5,6)$ | $-1$ |
$120$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.