# Properties

 Label 6.364871.7t7.a.a Dimension 6 Group $S_7$ Conductor $13^{2} \cdot 17 \cdot 127$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $6$ Group: $S_7$ Conductor: $364871= 13^{2} \cdot 17 \cdot 127$ Artin number field: Splitting field of 7.1.364871.1 defined by $f= x^{7} - x^{6} + 2 x^{4} + 2 x + 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $S_7$ Parity: Odd Determinant: 1.2159.2t1.a.a Projective image: $S_7$ Projective field: Galois closure of 7.1.364871.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 83 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 83 }$: $x^{2} + 82 x + 2$
Roots:
 $r_{ 1 }$ $=$ $9 a + 71 + \left(75 a + 82\right)\cdot 83 + \left(23 a + 70\right)\cdot 83^{2} + \left(28 a + 32\right)\cdot 83^{3} + \left(7 a + 37\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$ $r_{ 2 }$ $=$ $22 + 67\cdot 83 + 82\cdot 83^{2} + 33\cdot 83^{3} + 42\cdot 83^{4} +O\left(83^{ 5 }\right)$ $r_{ 3 }$ $=$ $82 a + 71 + \left(37 a + 64\right)\cdot 83 + \left(58 a + 26\right)\cdot 83^{2} + \left(67 a + 33\right)\cdot 83^{3} + \left(a + 6\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$ $r_{ 4 }$ $=$ $48 a + 27 + \left(11 a + 33\right)\cdot 83 + \left(27 a + 34\right)\cdot 83^{2} + \left(33 a + 31\right)\cdot 83^{3} + \left(56 a + 8\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$ $r_{ 5 }$ $=$ $a + 70 + \left(45 a + 20\right)\cdot 83 + \left(24 a + 47\right)\cdot 83^{2} + \left(15 a + 42\right)\cdot 83^{3} + \left(81 a + 23\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$ $r_{ 6 }$ $=$ $74 a + 80 + \left(7 a + 65\right)\cdot 83 + \left(59 a + 19\right)\cdot 83^{2} + \left(54 a + 37\right)\cdot 83^{3} + \left(75 a + 16\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$ $r_{ 7 }$ $=$ $35 a + 75 + \left(71 a + 79\right)\cdot 83 + \left(55 a + 49\right)\cdot 83^{2} + \left(49 a + 37\right)\cdot 83^{3} + \left(26 a + 31\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$

### 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.