# Properties

 Label 5.17923019113.6t14.b Dimension $5$ Group $S_5$ Conductor $17923019113$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $5$ Group: $S_5$ Conductor: $$17923019113$$$$\medspace = 2617^{3}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 5.1.2617.1 Galois orbit size: $1$ Smallest permutation container: $\PGL(2,5)$ Parity: even Projective image: $S_5$ Projective field: 5.1.2617.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $$x^{2} + 24 x + 2$$
Roots:
 $r_{ 1 }$ $=$ $$11 a + 13 + \left(16 a + 17\right)\cdot 29 + \left(12 a + 18\right)\cdot 29^{2} + \left(21 a + 28\right)\cdot 29^{3} + \left(4 a + 19\right)\cdot 29^{4} +O(29^{5})$$ $r_{ 2 }$ $=$ $$18 a + 10 + \left(12 a + 1\right)\cdot 29 + \left(16 a + 7\right)\cdot 29^{2} + \left(7 a + 7\right)\cdot 29^{3} + \left(24 a + 22\right)\cdot 29^{4} +O(29^{5})$$ $r_{ 3 }$ $=$ $$20 a + 20 + \left(10 a + 21\right)\cdot 29 + \left(27 a + 26\right)\cdot 29^{2} + \left(10 a + 10\right)\cdot 29^{3} + \left(14 a + 27\right)\cdot 29^{4} +O(29^{5})$$ $r_{ 4 }$ $=$ $$9 a + 4 + \left(18 a + 26\right)\cdot 29 + \left(a + 7\right)\cdot 29^{2} + \left(18 a + 9\right)\cdot 29^{3} + \left(14 a + 1\right)\cdot 29^{4} +O(29^{5})$$ $r_{ 5 }$ $=$ $$11 + 20\cdot 29 + 26\cdot 29^{2} + 29^{3} + 16\cdot 29^{4} +O(29^{5})$$

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