# Properties

 Label 6.382631.7t7.1c1 Dimension 6 Group $S_7$ Conductor $382631$ Root number 1 Frobenius-Schur indicator 1

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## Basic invariants

 Dimension: $6$ Group: $S_7$ Conductor: $382631$ Artin number field: Splitting field of $f= x^{7} - x^{5} + x^{3} - x - 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $S_7$ Parity: Odd Determinant: 1.382631.2t1.1c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 103 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 103 }$: $x^{2} + 102 x + 5$
Roots:
 $r_{ 1 }$ $=$ $2 a + 70 + \left(35 a + 89\right)\cdot 103 + \left(31 a + 99\right)\cdot 103^{2} + \left(24 a + 3\right)\cdot 103^{3} + \left(88 a + 70\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$ $r_{ 2 }$ $=$ $55 a + 3 + \left(50 a + 68\right)\cdot 103 + \left(39 a + 54\right)\cdot 103^{2} + \left(27 a + 21\right)\cdot 103^{3} + \left(21 a + 78\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$ $r_{ 3 }$ $=$ $48 a + 58 + \left(52 a + 63\right)\cdot 103 + \left(63 a + 43\right)\cdot 103^{2} + \left(75 a + 9\right)\cdot 103^{3} + \left(81 a + 72\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$ $r_{ 4 }$ $=$ $101 a + 72 + \left(67 a + 19\right)\cdot 103 + \left(71 a + 96\right)\cdot 103^{2} + \left(78 a + 99\right)\cdot 103^{3} + \left(14 a + 30\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$ $r_{ 5 }$ $=$ $19 + 14\cdot 103 + 53\cdot 103^{2} + 69\cdot 103^{3} + 50\cdot 103^{4} +O\left(103^{ 5 }\right)$ $r_{ 6 }$ $=$ $101 + 77\cdot 103 + 66\cdot 103^{2} + 92\cdot 103^{3} + 70\cdot 103^{4} +O\left(103^{ 5 }\right)$ $r_{ 7 }$ $=$ $89 + 78\cdot 103 + 100\cdot 103^{2} + 11\cdot 103^{3} + 39\cdot 103^{4} +O\left(103^{ 5 }\right)$

### 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.