# Properties

 Label 4.3e3_5e3_11e3_139e3.10t12.1c1 Dimension 4 Group $S_5$ Conductor $3^{3} \cdot 5^{3} \cdot 11^{3} \cdot 139^{3}$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $4$ Group: $S_5$ Conductor: $12064136250375= 3^{3} \cdot 5^{3} \cdot 11^{3} \cdot 139^{3}$ Artin number field: Splitting field of $f= x^{5} + x^{3} - 2 x^{2} - 4 x + 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $S_5$ Parity: Odd Determinant: 1.3_5_11_139.2t1.1c1

## 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 } &= 7 a + 4 + \left(11 a + 10\right)\cdot 13 + \left(2 a + 5\right)\cdot 13^{2} + \left(2 a + 5\right)\cdot 13^{3} + \left(a + 9\right)\cdot 13^{4} +O\left(13^{ 5 }\right) \\ r_{ 2 } &= 6 a + 11 + \left(a + 1\right)\cdot 13 + \left(10 a + 10\right)\cdot 13^{2} + \left(10 a + 4\right)\cdot 13^{3} + \left(11 a + 8\right)\cdot 13^{4} +O\left(13^{ 5 }\right) \\ r_{ 3 } &= 3 a + 5 + \left(12 a + 3\right)\cdot 13 + 10 a\cdot 13^{2} + \left(3 a + 8\right)\cdot 13^{3} + \left(10 a + 4\right)\cdot 13^{4} +O\left(13^{ 5 }\right) \\ r_{ 4 } &= 11 + 10\cdot 13 + 10\cdot 13^{2} + 6\cdot 13^{3} + 5\cdot 13^{4} +O\left(13^{ 5 }\right) \\ r_{ 5 } &= 10 a + 8 + 12\cdot 13 + \left(2 a + 11\right)\cdot 13^{2} + 9 a\cdot 13^{3} + \left(2 a + 11\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 $1$ $1$ $()$ $4$ $10$ $2$ $(1,2)$ $-2$ $15$ $2$ $(1,2)(3,4)$ $0$ $20$ $3$ $(1,2,3)$ $1$ $30$ $4$ $(1,2,3,4)$ $0$ $24$ $5$ $(1,2,3,4,5)$ $-1$ $20$ $6$ $(1,2,3)(4,5)$ $1$
The blue line marks the conjugacy class containing complex conjugation.