# Properties

 Label 6.11_41_461.7t7.1c1 Dimension 6 Group $S_7$ Conductor $11 \cdot 41 \cdot 461$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $6$ Group: $S_7$ Conductor: $207911= 11 \cdot 41 \cdot 461$ Artin number field: Splitting field of $f= x^{7} - x^{5} - x^{4} - x^{3} + x^{2} + x + 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $S_7$ Parity: Odd Determinant: 1.11_41_461.2t1.1c1

## 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:
 $r_{ 1 }$ $=$ $44 a + 152 + \left(119 a + 131\right)\cdot 157 + \left(34 a + 89\right)\cdot 157^{2} + \left(123 a + 101\right)\cdot 157^{3} + \left(112 a + 63\right)\cdot 157^{4} +O\left(157^{ 5 }\right)$ $r_{ 2 }$ $=$ $53 a + 74 + \left(62 a + 109\right)\cdot 157 + \left(5 a + 122\right)\cdot 157^{2} + \left(80 a + 49\right)\cdot 157^{3} + \left(81 a + 67\right)\cdot 157^{4} +O\left(157^{ 5 }\right)$ $r_{ 3 }$ $=$ $134 a + 31 + \left(82 a + 25\right)\cdot 157 + \left(152 a + 20\right)\cdot 157^{2} + \left(124 a + 156\right)\cdot 157^{3} + \left(133 a + 125\right)\cdot 157^{4} +O\left(157^{ 5 }\right)$ $r_{ 4 }$ $=$ $104 a + 25 + \left(94 a + 54\right)\cdot 157 + \left(151 a + 87\right)\cdot 157^{2} + \left(76 a + 130\right)\cdot 157^{3} + \left(75 a + 80\right)\cdot 157^{4} +O\left(157^{ 5 }\right)$ $r_{ 5 }$ $=$ $58 + 102\cdot 157 + 91\cdot 157^{2} + 134\cdot 157^{3} + 57\cdot 157^{4} +O\left(157^{ 5 }\right)$ $r_{ 6 }$ $=$ $23 a + 73 + \left(74 a + 148\right)\cdot 157 + \left(4 a + 71\right)\cdot 157^{2} + 32 a\cdot 157^{3} + \left(23 a + 42\right)\cdot 157^{4} +O\left(157^{ 5 }\right)$ $r_{ 7 }$ $=$ $113 a + 58 + \left(37 a + 56\right)\cdot 157 + \left(122 a + 144\right)\cdot 157^{2} + \left(33 a + 54\right)\cdot 157^{3} + \left(44 a + 33\right)\cdot 157^{4} +O\left(157^{ 5 }\right)$

### Generators of the action on the roots $r_1, \ldots, r_{ 7 }$

 Cycle notation $(1,2,3,4,5,6,7)$ $(1,2)$

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 7 }$ Character value $1$ $1$ $()$ $6$ $21$ $2$ $(1,2)$ $4$ $105$ $2$ $(1,2)(3,4)(5,6)$ $0$ $105$ $2$ $(1,2)(3,4)$ $2$ $70$ $3$ $(1,2,3)$ $3$ $280$ $3$ $(1,2,3)(4,5,6)$ $0$ $210$ $4$ $(1,2,3,4)$ $2$ $630$ $4$ $(1,2,3,4)(5,6)$ $0$ $504$ $5$ $(1,2,3,4,5)$ $1$ $210$ $6$ $(1,2,3)(4,5)(6,7)$ $-1$ $420$ $6$ $(1,2,3)(4,5)$ $1$ $840$ $6$ $(1,2,3,4,5,6)$ $0$ $720$ $7$ $(1,2,3,4,5,6,7)$ $-1$ $504$ $10$ $(1,2,3,4,5)(6,7)$ $-1$ $420$ $12$ $(1,2,3,4)(5,6,7)$ $-1$
The blue line marks the conjugacy class containing complex conjugation.