# Properties

 Label 5.7e3_25183e3.6t14.1c1 Dimension 5 Group $S_5$ Conductor $7^{3} \cdot 25183^{3}$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $5$ Group: $S_5$ Conductor: $5477930481596041= 7^{3} \cdot 25183^{3}$ Artin number field: Splitting field of $f= x^{5} - x^{4} - 5 x^{3} + 3 x^{2} + 4 x - 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $\PGL(2,5)$ Parity: Even Determinant: 1.7_25183.2t1.1c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: $x^{2} + 12 x + 2$
Roots: \begin{aligned} r_{ 1 } &= 10 a + 4 + 12\cdot 13 + \left(a + 7\right)\cdot 13^{2} + 5\cdot 13^{3} + \left(12 a + 12\right)\cdot 13^{4} +O\left(13^{ 5 }\right) \\ r_{ 2 } &= a + 9 + \left(5 a + 7\right)\cdot 13 + \left(a + 11\right)\cdot 13^{2} + \left(7 a + 3\right)\cdot 13^{3} + \left(11 a + 5\right)\cdot 13^{4} +O\left(13^{ 5 }\right) \\ r_{ 3 } &= 3 + 4\cdot 13 + 3\cdot 13^{2} + 2\cdot 13^{3} +O\left(13^{ 5 }\right) \\ r_{ 4 } &= 3 a + 1 + \left(12 a + 3\right)\cdot 13 + \left(11 a + 8\right)\cdot 13^{2} + \left(12 a + 4\right)\cdot 13^{3} + 11\cdot 13^{4} +O\left(13^{ 5 }\right) \\ r_{ 5 } &= 12 a + 10 + \left(7 a + 11\right)\cdot 13 + \left(11 a + 7\right)\cdot 13^{2} + \left(5 a + 9\right)\cdot 13^{3} + \left(a + 9\right)\cdot 13^{4} +O\left(13^{ 5 }\right) \\ \end{aligned}

### 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.