# Properties

 Label 6.242147.7t7.a.a Dimension 6 Group $S_7$ Conductor $242147$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $6$ Group: $S_7$ Conductor: $242147$ Artin number field: Splitting field of 7.1.242147.1 defined by $f= x^{7} - 2 x^{6} + 2 x^{5} - x^{4} - x^{3} + 2 x^{2} - x + 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $S_7$ Parity: Odd Determinant: 1.242147.2t1.a.a Projective image: $S_7$ Projective field: Galois closure of 7.1.242147.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 89 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 89 }$: $x^{2} + 82 x + 3$
Roots:
 $r_{ 1 }$ $=$ $28 + 20\cdot 89 + 50\cdot 89^{2} + 13\cdot 89^{3} + 64\cdot 89^{4} +O\left(89^{ 5 }\right)$ $r_{ 2 }$ $=$ $67 a + 27 + \left(5 a + 4\right)\cdot 89 + \left(52 a + 6\right)\cdot 89^{2} + \left(38 a + 84\right)\cdot 89^{3} + \left(32 a + 60\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$ $r_{ 3 }$ $=$ $82 a + \left(41 a + 25\right)\cdot 89 + \left(23 a + 55\right)\cdot 89^{2} + \left(26 a + 58\right)\cdot 89^{3} + \left(23 a + 22\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$ $r_{ 4 }$ $=$ $7 a + 40 + \left(47 a + 58\right)\cdot 89 + \left(65 a + 88\right)\cdot 89^{2} + \left(62 a + 40\right)\cdot 89^{3} + \left(65 a + 70\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$ $r_{ 5 }$ $=$ $22 a + 51 + \left(83 a + 66\right)\cdot 89 + \left(36 a + 8\right)\cdot 89^{2} + \left(50 a + 35\right)\cdot 89^{3} + \left(56 a + 71\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$ $r_{ 6 }$ $=$ $29 + 13\cdot 89 + 18\cdot 89^{2} + 11\cdot 89^{3} + 83\cdot 89^{4} +O\left(89^{ 5 }\right)$ $r_{ 7 }$ $=$ $5 + 79\cdot 89 + 39\cdot 89^{2} + 23\cdot 89^{3} + 72\cdot 89^{4} +O\left(89^{ 5 }\right)$

### 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.