# Properties

 Label 10.54419558400000000.30t164.c.a Dimension 10 Group $S_6$ Conductor $2^{18} \cdot 3^{12} \cdot 5^{8}$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $10$ Group: $S_6$ Conductor: $54419558400000000= 2^{18} \cdot 3^{12} \cdot 5^{8}$ Artin number field: Splitting field of 6.2.648000.1 defined by $f= x^{6} + 2 x^{4} - 2 x^{3} - 3 x^{2} - 6 x - 3$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: 30T164 Parity: Even Determinant: 1.1.1t1.a.a Projective image: $S_6$ Projective field: Galois closure of 6.2.648000.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 137 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 137 }$: $x^{2} + 131 x + 3$
Roots:
 $r_{ 1 }$ $=$ $41 a + 70 + \left(43 a + 31\right)\cdot 137 + \left(73 a + 12\right)\cdot 137^{2} + \left(136 a + 9\right)\cdot 137^{3} + \left(94 a + 98\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$ $r_{ 2 }$ $=$ $116 a + 57 + \left(90 a + 11\right)\cdot 137 + \left(14 a + 57\right)\cdot 137^{2} + \left(99 a + 99\right)\cdot 137^{3} + \left(66 a + 19\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$ $r_{ 3 }$ $=$ $102 a + 55 + \left(101 a + 132\right)\cdot 137 + \left(21 a + 61\right)\cdot 137^{2} + \left(60 a + 86\right)\cdot 137^{3} + \left(44 a + 96\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$ $r_{ 4 }$ $=$ $35 a + 119 + \left(35 a + 92\right)\cdot 137 + \left(115 a + 90\right)\cdot 137^{2} + \left(76 a + 14\right)\cdot 137^{3} + \left(92 a + 29\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$ $r_{ 5 }$ $=$ $96 a + 42 + \left(93 a + 113\right)\cdot 137 + \left(63 a + 134\right)\cdot 137^{2} + 69\cdot 137^{3} + \left(42 a + 120\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$ $r_{ 6 }$ $=$ $21 a + 68 + \left(46 a + 29\right)\cdot 137 + \left(122 a + 54\right)\cdot 137^{2} + \left(37 a + 131\right)\cdot 137^{3} + \left(70 a + 46\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$

### Generators of the action on the roots $r_1, \ldots, r_{ 6 }$

 Cycle notation $(1,2)$ $(1,2,3,4,5,6)$

### 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.