# Properties

 Label 9.1719926784000000.20t145.b.a Dimension 9 Group $S_6$ Conductor $2^{24} \cdot 3^{8} \cdot 5^{6}$ Root number 1 Frobenius-Schur indicator 1

# Learn more about

## Basic invariants

 Dimension: $9$ Group: $S_6$ Conductor: $1719926784000000= 2^{24} \cdot 3^{8} \cdot 5^{6}$ Artin number field: Splitting field of $f= x^{6} - 2 x^{5} - x^{4} + 6 x^{3} - 2 x^{2} - 4 x - 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: 20T145 Parity: Even Determinant: 1.1.1t1.a.a

## 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:
 $r_{ 1 }$ $=$ $118 a + 5 + \left(93 a + 77\right)\cdot 137 + \left(97 a + 36\right)\cdot 137^{2} + \left(50 a + 102\right)\cdot 137^{3} + \left(121 a + 19\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$ $r_{ 2 }$ $=$ $19 a + 28 + \left(43 a + 111\right)\cdot 137 + \left(39 a + 117\right)\cdot 137^{2} + \left(86 a + 34\right)\cdot 137^{3} + \left(15 a + 12\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$ $r_{ 3 }$ $=$ $51 + 99\cdot 137 + 51\cdot 137^{2} + 46\cdot 137^{3} + 97\cdot 137^{4} +O\left(137^{ 5 }\right)$ $r_{ 4 }$ $=$ $29 a + 88 + \left(21 a + 131\right)\cdot 137 + \left(51 a + 24\right)\cdot 137^{2} + \left(96 a + 68\right)\cdot 137^{3} + \left(91 a + 92\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$ $r_{ 5 }$ $=$ $116 + 35\cdot 137 + 6\cdot 137^{2} + 112\cdot 137^{3} + 53\cdot 137^{4} +O\left(137^{ 5 }\right)$ $r_{ 6 }$ $=$ $108 a + 125 + \left(115 a + 92\right)\cdot 137 + \left(85 a + 36\right)\cdot 137^{2} + \left(40 a + 47\right)\cdot 137^{3} + \left(45 a + 135\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$

### 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 $1$ $1$ $()$ $9$ $15$ $2$ $(1,2)(3,4)(5,6)$ $-3$ $15$ $2$ $(1,2)$ $-3$ $45$ $2$ $(1,2)(3,4)$ $1$ $40$ $3$ $(1,2,3)(4,5,6)$ $0$ $40$ $3$ $(1,2,3)$ $0$ $90$ $4$ $(1,2,3,4)(5,6)$ $1$ $90$ $4$ $(1,2,3,4)$ $1$ $144$ $5$ $(1,2,3,4,5)$ $-1$ $120$ $6$ $(1,2,3,4,5,6)$ $0$ $120$ $6$ $(1,2,3)(4,5)$ $0$
The blue line marks the conjugacy class containing complex conjugation.