# Properties

 Label 6.13_29_991.7t7.1c1 Dimension 6 Group $S_7$ Conductor $13 \cdot 29 \cdot 991$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $6$ Group: $S_7$ Conductor: $373607= 13 \cdot 29 \cdot 991$ Artin number field: Splitting field of $f= x^{7} - x^{6} + x^{5} - 2 x^{3} + 3 x^{2} - 2 x + 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $S_7$ Parity: Odd Determinant: 1.13_29_991.2t1.1c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: $x^{2} + 21 x + 5$
Roots:
 $r_{ 1 }$ $=$ $4 + 23^{2} + 22\cdot 23^{3} + 22\cdot 23^{4} +O\left(23^{ 5 }\right)$ $r_{ 2 }$ $=$ $3 a + 16 + \left(2 a + 4\right)\cdot 23 + \left(18 a + 4\right)\cdot 23^{2} + \left(17 a + 4\right)\cdot 23^{3} + \left(10 a + 20\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$ $r_{ 3 }$ $=$ $20 a + 22 + \left(20 a + 5\right)\cdot 23 + \left(4 a + 15\right)\cdot 23^{2} + \left(5 a + 21\right)\cdot 23^{3} + 12 a\cdot 23^{4} +O\left(23^{ 5 }\right)$ $r_{ 4 }$ $=$ $10 a + 15 + \left(12 a + 13\right)\cdot 23 + \left(a + 4\right)\cdot 23^{2} + \left(10 a + 13\right)\cdot 23^{3} + \left(8 a + 11\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$ $r_{ 5 }$ $=$ $13 a + 12 + \left(10 a + 5\right)\cdot 23 + \left(21 a + 18\right)\cdot 23^{2} + \left(12 a + 8\right)\cdot 23^{3} + \left(14 a + 18\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$ $r_{ 6 }$ $=$ $12 a + \left(21 a + 4\right)\cdot 23 + \left(13 a + 21\right)\cdot 23^{2} + \left(2 a + 3\right)\cdot 23^{3} + \left(7 a + 3\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$ $r_{ 7 }$ $=$ $11 a + 1 + \left(a + 12\right)\cdot 23 + \left(9 a + 4\right)\cdot 23^{2} + \left(20 a + 18\right)\cdot 23^{3} + \left(15 a + 14\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$

### 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.