# Properties

 Label 10.12548716987355136.30t164.b.a Dimension 10 Group $S_6$ Conductor $2^{12} \cdot 3^{12} \cdot 7^{8}$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $10$ Group: $S_6$ Conductor: $12548716987355136= 2^{12} \cdot 3^{12} \cdot 7^{8}$ Artin number field: Splitting field of 6.0.12446784.1 defined by $f= x^{6} + 5 x^{4} - 2 x^{3} + 9 x^{2} - 8 x + 16$ 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 6.0.12446784.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 157 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 157 }$: $x^{2} + 152 x + 5$
Roots:
 $r_{ 1 }$ $=$ $90 a + 84 + \left(66 a + 4\right)\cdot 157 + \left(81 a + 121\right)\cdot 157^{2} + \left(156 a + 37\right)\cdot 157^{3} + \left(15 a + 98\right)\cdot 157^{4} +O\left(157^{ 5 }\right)$ $r_{ 2 }$ $=$ $a + 124 + \left(3 a + 52\right)\cdot 157 + \left(154 a + 33\right)\cdot 157^{2} + \left(62 a + 31\right)\cdot 157^{3} + \left(140 a + 143\right)\cdot 157^{4} +O\left(157^{ 5 }\right)$ $r_{ 3 }$ $=$ $67 a + 63 + \left(90 a + 90\right)\cdot 157 + \left(75 a + 147\right)\cdot 157^{2} + 110\cdot 157^{3} + \left(141 a + 21\right)\cdot 157^{4} +O\left(157^{ 5 }\right)$ $r_{ 4 }$ $=$ $156 a + 129 + \left(153 a + 66\right)\cdot 157 + \left(2 a + 15\right)\cdot 157^{2} + \left(94 a + 35\right)\cdot 157^{3} + \left(16 a + 154\right)\cdot 157^{4} +O\left(157^{ 5 }\right)$ $r_{ 5 }$ $=$ $43 + 58\cdot 157 + 29\cdot 157^{2} + 157^{3} + 61\cdot 157^{4} +O\left(157^{ 5 }\right)$ $r_{ 6 }$ $=$ $28 + 41\cdot 157 + 124\cdot 157^{2} + 97\cdot 157^{3} + 149\cdot 157^{4} +O\left(157^{ 5 }\right)$

### 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.