# Properties

 Label 5.58492928.6t14.d.a Dimension $5$ Group $\PGL(2,5)$ Conductor $58492928$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $5$ Group: $\PGL(2,5)$ Conductor: $$58492928$$$$\medspace = 2^{11} \cdot 13^{4}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 6.0.58492928.4 Galois orbit size: $1$ Smallest permutation container: $\PGL(2,5)$ Parity: odd Determinant: 1.8.2t1.b.a Projective image: $S_5$ Projective stem field: Galois closure of 6.0.58492928.4

## Defining polynomial

 $f(x)$ $=$ $$x^{6} - x^{5} - 3x^{4} - 6x^{3} - x^{2} + 23x + 27$$ x^6 - x^5 - 3*x^4 - 6*x^3 - x^2 + 23*x + 27 .

The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: $$x^{2} + 18x + 2$$

Roots:
 $r_{ 1 }$ $=$ $$17 a + 15 + \left(8 a + 11\right)\cdot 19 + \left(17 a + 11\right)\cdot 19^{2} + \left(7 a + 9\right)\cdot 19^{3} + \left(13 a + 7\right)\cdot 19^{4} +O(19^{5})$$ 17*a + 15 + (8*a + 11)*19 + (17*a + 11)*19^2 + (7*a + 9)*19^3 + (13*a + 7)*19^4+O(19^5) $r_{ 2 }$ $=$ $$2 a + 13 + \left(10 a + 3\right)\cdot 19 + \left(a + 1\right)\cdot 19^{2} + 11 a\cdot 19^{3} + \left(5 a + 13\right)\cdot 19^{4} +O(19^{5})$$ 2*a + 13 + (10*a + 3)*19 + (a + 1)*19^2 + 11*a*19^3 + (5*a + 13)*19^4+O(19^5) $r_{ 3 }$ $=$ $$15 + 6\cdot 19 + 2\cdot 19^{2} + 6\cdot 19^{3} + 10\cdot 19^{4} +O(19^{5})$$ 15 + 6*19 + 2*19^2 + 6*19^3 + 10*19^4+O(19^5) $r_{ 4 }$ $=$ $$11 + 19 + 14\cdot 19^{3} + 3\cdot 19^{4} +O(19^{5})$$ 11 + 19 + 14*19^3 + 3*19^4+O(19^5) $r_{ 5 }$ $=$ $$16 a + 13 + \left(3 a + 3\right)\cdot 19 + \left(11 a + 17\right)\cdot 19^{2} + \left(11 a + 3\right)\cdot 19^{3} + \left(2 a + 6\right)\cdot 19^{4} +O(19^{5})$$ 16*a + 13 + (3*a + 3)*19 + (11*a + 17)*19^2 + (11*a + 3)*19^3 + (2*a + 6)*19^4+O(19^5) $r_{ 6 }$ $=$ $$3 a + 10 + \left(15 a + 10\right)\cdot 19 + \left(7 a + 5\right)\cdot 19^{2} + \left(7 a + 4\right)\cdot 19^{3} + \left(16 a + 16\right)\cdot 19^{4} +O(19^{5})$$ 3*a + 10 + (15*a + 10)*19 + (7*a + 5)*19^2 + (7*a + 4)*19^3 + (16*a + 16)*19^4+O(19^5)

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

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

## Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 6 }$ Character value $1$ $1$ $()$ $5$ $10$ $2$ $(1,5)(2,6)(3,4)$ $-1$ $15$ $2$ $(1,2)(5,6)$ $1$ $20$ $3$ $(1,2,5)(3,4,6)$ $-1$ $30$ $4$ $(1,6,4,2)$ $1$ $24$ $5$ $(1,4,5,6,3)$ $0$ $20$ $6$ $(1,6,2,3,5,4)$ $-1$

The blue line marks the conjugacy class containing complex conjugation.