# Properties

 Label 6.83_3037.7t7.1c1 Dimension 6 Group $S_7$ Conductor $83 \cdot 3037$ Root number 1 Frobenius-Schur indicator 1

# Learn more about

## Basic invariants

 Dimension: $6$ Group: $S_7$ Conductor: $252071= 83 \cdot 3037$ Artin number field: Splitting field of $f= x^{7} - x^{6} + 2 x^{4} - x^{3} - 2 x^{2} + x + 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $S_7$ Parity: Odd Determinant: 1.83_3037.2t1.1c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 167 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 167 }$: $x^{2} + 166 x + 5$
Roots:
 $r_{ 1 }$ $=$ $144 + 16\cdot 167 + 146\cdot 167^{2} + 130\cdot 167^{3} + 61\cdot 167^{4} +O\left(167^{ 5 }\right)$ $r_{ 2 }$ $=$ $58 + 84\cdot 167 + 31\cdot 167^{2} + 68\cdot 167^{3} + 89\cdot 167^{4} +O\left(167^{ 5 }\right)$ $r_{ 3 }$ $=$ $83 a + 16 + \left(72 a + 80\right)\cdot 167 + \left(71 a + 93\right)\cdot 167^{2} + \left(14 a + 104\right)\cdot 167^{3} + \left(48 a + 10\right)\cdot 167^{4} +O\left(167^{ 5 }\right)$ $r_{ 4 }$ $=$ $35 a + 41 + \left(126 a + 134\right)\cdot 167 + \left(62 a + 7\right)\cdot 167^{2} + \left(129 a + 10\right)\cdot 167^{3} + \left(41 a + 148\right)\cdot 167^{4} +O\left(167^{ 5 }\right)$ $r_{ 5 }$ $=$ $84 a + 99 + \left(94 a + 69\right)\cdot 167 + \left(95 a + 92\right)\cdot 167^{2} + \left(152 a + 47\right)\cdot 167^{3} + \left(118 a + 44\right)\cdot 167^{4} +O\left(167^{ 5 }\right)$ $r_{ 6 }$ $=$ $132 a + 76 + \left(40 a + 58\right)\cdot 167 + \left(104 a + 111\right)\cdot 167^{2} + \left(37 a + 76\right)\cdot 167^{3} + \left(125 a + 60\right)\cdot 167^{4} +O\left(167^{ 5 }\right)$ $r_{ 7 }$ $=$ $68 + 57\cdot 167 + 18\cdot 167^{2} + 63\cdot 167^{3} + 86\cdot 167^{4} +O\left(167^{ 5 }\right)$

### 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.