# Properties

 Label 6.202471.7t7.a.a Dimension $6$ Group $S_7$ Conductor $202471$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $6$ Group: $S_7$ Conductor: $$202471$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 7.1.202471.1 Galois orbit size: $1$ Smallest permutation container: $S_7$ Parity: odd Determinant: 1.202471.2t1.a.a Projective image: $S_7$ Projective field: Galois closure of 7.1.202471.1

## Defining polynomial

 $f(x)$ $=$ $x^{7} - x^{6} - x^{5} + 2 x^{4} - x^{2} + 1$.

The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: $x^{2} + 29 x + 3$

Roots:
 $r_{ 1 }$ $=$ $8 + 2\cdot 31 + 16\cdot 31^{2} + 19\cdot 31^{3} + 14\cdot 31^{4} +O\left(31^{ 5 }\right)$ $r_{ 2 }$ $=$ $12 a + 3 + \left(a + 2\right)\cdot 31 + 30 a\cdot 31^{2} + \left(25 a + 19\right)\cdot 31^{3} + \left(8 a + 23\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ $r_{ 3 }$ $=$ $4 a + 3 + \left(17 a + 25\right)\cdot 31 + \left(30 a + 4\right)\cdot 31^{2} + \left(18 a + 12\right)\cdot 31^{3} + \left(8 a + 25\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ $r_{ 4 }$ $=$ $27 a + 11 + \left(13 a + 24\right)\cdot 31 + 17\cdot 31^{2} + \left(12 a + 19\right)\cdot 31^{3} + \left(22 a + 23\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ $r_{ 5 }$ $=$ $9 a + 12 + \left(30 a + 28\right)\cdot 31 + \left(24 a + 18\right)\cdot 31^{2} + \left(22 a + 11\right)\cdot 31^{3} + \left(16 a + 5\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ $r_{ 6 }$ $=$ $19 a + 27 + \left(29 a + 23\right)\cdot 31 + 27\cdot 31^{2} + \left(5 a + 9\right)\cdot 31^{3} + \left(22 a + 15\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ $r_{ 7 }$ $=$ $22 a + 30 + 17\cdot 31 + \left(6 a + 7\right)\cdot 31^{2} + \left(8 a + 1\right)\cdot 31^{3} + \left(14 a + 16\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$

## 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.