# Properties

 Label 4.11e2_37.5t5.1 Dimension 4 Group $S_5$ Conductor $11^{2} \cdot 37$ Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $4$ Group: $S_5$ Conductor: $4477= 11^{2} \cdot 37$ Artin number field: Splitting field of $f= x^{5} + x^{3} + x - 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $S_5$ Parity: Even

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 101 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 101 }$: $x^{2} + 97 x + 2$
Roots:
 $r_{ 1 }$ $=$ $90 a + 50 + \left(29 a + 88\right)\cdot 101 + \left(89 a + 37\right)\cdot 101^{2} + \left(83 a + 87\right)\cdot 101^{3} + \left(36 a + 82\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$ $r_{ 2 }$ $=$ $11 a + 6 + \left(71 a + 17\right)\cdot 101 + \left(11 a + 62\right)\cdot 101^{2} + \left(17 a + 30\right)\cdot 101^{3} + \left(64 a + 45\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$ $r_{ 3 }$ $=$ $48 + 31\cdot 101 + 100\cdot 101^{2} + 32\cdot 101^{3} + 5\cdot 101^{4} +O\left(101^{ 5 }\right)$ $r_{ 4 }$ $=$ $72 a + 6 + \left(42 a + 84\right)\cdot 101 + \left(46 a + 80\right)\cdot 101^{2} + \left(83 a + 83\right)\cdot 101^{3} + \left(76 a + 73\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$ $r_{ 5 }$ $=$ $29 a + 92 + \left(58 a + 81\right)\cdot 101 + \left(54 a + 21\right)\cdot 101^{2} + \left(17 a + 68\right)\cdot 101^{3} + \left(24 a + 95\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$

### 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 values $c1$ $1$ $1$ $()$ $4$ $10$ $2$ $(1,2)$ $2$ $15$ $2$ $(1,2)(3,4)$ $0$ $20$ $3$ $(1,2,3)$ $1$ $30$ $4$ $(1,2,3,4)$ $0$ $24$ $5$ $(1,2,3,4,5)$ $-1$ $20$ $6$ $(1,2,3)(4,5)$ $-1$
The blue line marks the conjugacy class containing complex conjugation.