# Properties

 Label 6.17923019113.20t30.a Dimension $6$ Group $\PGL(2,5)$ Conductor $17923019113$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $6$ Group: $\PGL(2,5)$ Conductor: $$17923019113$$$$\medspace = 2617^{3}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 6.2.17923019113.1 Galois orbit size: $1$ Smallest permutation container: 20T30 Parity: even Projective image: $S_5$ Projective field: 6.2.17923019113.1

## 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:
 $r_{ 1 }$ $=$ $$3 + 23\cdot 29 + 27\cdot 29^{3} + 2\cdot 29^{4} +O(29^{5})$$ $r_{ 2 }$ $=$ $$12 + 29 + 21\cdot 29^{2} + 4\cdot 29^{3} + 10\cdot 29^{4} +O(29^{5})$$ $r_{ 3 }$ $=$ $$9 a + 15 + \left(23 a + 15\right)\cdot 29 + \left(3 a + 1\right)\cdot 29^{2} + \left(11 a + 7\right)\cdot 29^{3} + \left(4 a + 20\right)\cdot 29^{4} +O(29^{5})$$ $r_{ 4 }$ $=$ $$20 a + 2 + \left(5 a + 7\right)\cdot 29 + \left(25 a + 26\right)\cdot 29^{2} + 17 a\cdot 29^{3} + \left(24 a + 2\right)\cdot 29^{4} +O(29^{5})$$ $r_{ 5 }$ $=$ $$17 a + \left(24 a + 25\right)\cdot 29 + \left(25 a + 9\right)\cdot 29^{2} + \left(28 a + 22\right)\cdot 29^{3} + \left(8 a + 17\right)\cdot 29^{4} +O(29^{5})$$ $r_{ 6 }$ $=$ $$12 a + 27 + \left(4 a + 14\right)\cdot 29 + \left(3 a + 27\right)\cdot 29^{2} + 24\cdot 29^{3} + \left(20 a + 4\right)\cdot 29^{4} +O(29^{5})$$

### Generators of the action on the roots $r_1, \ldots, r_{ 6 }$

 Cycle notation $(1,6,2,4,5,3)$ $(1,5)(2,4)(3,6)$

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 6 }$ Character values $c1$ $1$ $1$ $()$ $6$ $10$ $2$ $(1,5)(2,4)(3,6)$ $0$ $15$ $2$ $(2,3)(4,6)$ $-2$ $20$ $3$ $(1,2,5)(3,6,4)$ $0$ $30$ $4$ $(1,4,6,2)$ $0$ $24$ $5$ $(1,3,2,5,6)$ $1$ $20$ $6$ $(1,6,2,4,5,3)$ $0$
The blue line marks the conjugacy class containing complex conjugation.