# Properties

 Label 5.48615135735473.6t14.a.a Dimension 5 Group $S_5$ Conductor $36497^{3}$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $5$ Group: $S_5$ Conductor: $48615135735473= 36497^{3}$ Artin number field: Splitting field of 5.5.36497.1 defined by $f= x^{5} - 2 x^{4} - 3 x^{3} + 5 x^{2} + x - 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $\PGL(2,5)$ Parity: Even Determinant: 1.36497.2t1.a.a Projective image: $S_5$ Projective field: Galois closure of 5.5.36497.1

## Galois action

### Roots of defining polynomial

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:
 $r_{ 1 }$ $=$ $5 a + 3 + \left(4 a + 27\right)\cdot 31 + \left(11 a + 28\right)\cdot 31^{2} + \left(19 a + 28\right)\cdot 31^{3} + \left(11 a + 8\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ $r_{ 2 }$ $=$ $7 a + 15 + \left(18 a + 24\right)\cdot 31 + \left(12 a + 2\right)\cdot 31^{2} + \left(6 a + 1\right)\cdot 31^{3} + \left(28 a + 18\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ $r_{ 3 }$ $=$ $26 a + 13 + \left(26 a + 30\right)\cdot 31 + \left(19 a + 15\right)\cdot 31^{2} + \left(11 a + 25\right)\cdot 31^{3} + \left(19 a + 12\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ $r_{ 4 }$ $=$ $4 + 19\cdot 31 + 4\cdot 31^{2} + 5\cdot 31^{3} + 16\cdot 31^{4} +O\left(31^{ 5 }\right)$ $r_{ 5 }$ $=$ $24 a + 29 + \left(12 a + 22\right)\cdot 31 + \left(18 a + 9\right)\cdot 31^{2} + \left(24 a + 1\right)\cdot 31^{3} + \left(2 a + 6\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$

### Generators of the action on the roots $r_1, \ldots, r_{ 5 }$

 Cycle notation $(1,2)$ $(1,2,3,4,5)$

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 5 }$ Character value $1$ $1$ $()$ $5$ $10$ $2$ $(1,2)$ $-1$ $15$ $2$ $(1,2)(3,4)$ $1$ $20$ $3$ $(1,2,3)$ $-1$ $30$ $4$ $(1,2,3,4)$ $1$ $24$ $5$ $(1,2,3,4,5)$ $0$ $20$ $6$ $(1,2,3)(4,5)$ $-1$
The blue line marks the conjugacy class containing complex conjugation.