# Properties

 Label 6.11_97_373.7t7.1c1 Dimension 6 Group $S_7$ Conductor $11 \cdot 97 \cdot 373$ Root number 1 Frobenius-Schur indicator 1

# Learn more about

## Basic invariants

 Dimension: $6$ Group: $S_7$ Conductor: $397991= 11 \cdot 97 \cdot 373$ Artin number field: Splitting field of $f= x^{7} - 2 x^{5} + 3 x^{3} - x^{2} - x + 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $S_7$ Parity: Odd Determinant: 1.11_97_373.2t1.1c1

## 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:
 $r_{ 1 }$ $=$ $2 + 32\cdot 83 + 6\cdot 83^{2} + 41\cdot 83^{3} + 26\cdot 83^{4} +O\left(83^{ 5 }\right)$ $r_{ 2 }$ $=$ $37 + 31\cdot 83 + 6\cdot 83^{2} + 37\cdot 83^{3} + 10\cdot 83^{4} +O\left(83^{ 5 }\right)$ $r_{ 3 }$ $=$ $79 a + 76 + 32\cdot 83 + \left(52 a + 63\right)\cdot 83^{2} + \left(27 a + 23\right)\cdot 83^{3} + \left(63 a + 16\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$ $r_{ 4 }$ $=$ $4 a + 72 + \left(82 a + 37\right)\cdot 83 + \left(30 a + 31\right)\cdot 83^{2} + \left(55 a + 82\right)\cdot 83^{3} + \left(19 a + 51\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$ $r_{ 5 }$ $=$ $8 a + 76 + \left(51 a + 65\right)\cdot 83 + \left(45 a + 24\right)\cdot 83^{2} + \left(52 a + 71\right)\cdot 83^{3} + \left(18 a + 4\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$ $r_{ 6 }$ $=$ $68 + 22\cdot 83 + 14\cdot 83^{2} + 81\cdot 83^{3} + 83^{4} +O\left(83^{ 5 }\right)$ $r_{ 7 }$ $=$ $75 a + 1 + \left(31 a + 26\right)\cdot 83 + \left(37 a + 19\right)\cdot 83^{2} + \left(30 a + 78\right)\cdot 83^{3} + \left(64 a + 53\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$

### 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.