# Properties

 Label 4.79e3_89e3.10t12.1c1 Dimension 4 Group $S_5$ Conductor $79^{3} \cdot 89^{3}$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $4$ Group: $S_5$ Conductor: $347577210791= 79^{3} \cdot 89^{3}$ Artin number field: Splitting field of $f= x^{5} - x^{3} - x^{2} - x + 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $S_5$ Parity: Odd Determinant: 1.79_89.2t1.1c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 59 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 59 }$: $x^{2} + 58 x + 2$
Roots: \begin{aligned} r_{ 1 } &= 57 a + 50 + \left(35 a + 28\right)\cdot 59 + \left(49 a + 10\right)\cdot 59^{2} + \left(49 a + 53\right)\cdot 59^{3} + \left(17 a + 56\right)\cdot 59^{4} +O\left(59^{ 5 }\right) \\ r_{ 2 } &= 23 + 15\cdot 59 + 46\cdot 59^{2} + 53\cdot 59^{3} + 35\cdot 59^{4} +O\left(59^{ 5 }\right) \\ r_{ 3 } &= 21 a + 47 + \left(52 a + 46\right)\cdot 59 + \left(45 a + 21\right)\cdot 59^{2} + \left(55 a + 3\right)\cdot 59^{3} + \left(57 a + 58\right)\cdot 59^{4} +O\left(59^{ 5 }\right) \\ r_{ 4 } &= 2 a + 48 + \left(23 a + 7\right)\cdot 59 + \left(9 a + 24\right)\cdot 59^{2} + \left(9 a + 53\right)\cdot 59^{3} + \left(41 a + 24\right)\cdot 59^{4} +O\left(59^{ 5 }\right) \\ r_{ 5 } &= 38 a + 9 + \left(6 a + 19\right)\cdot 59 + \left(13 a + 15\right)\cdot 59^{2} + \left(3 a + 13\right)\cdot 59^{3} + \left(a + 1\right)\cdot 59^{4} +O\left(59^{ 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.