# Properties

 Label 4.180649580077.10t12.a.a Dimension $4$ Group $S_5$ Conductor $180649580077$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $4$ Group: $S_5$ Conductor: $$180649580077$$$$\medspace = 5653^{3}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 5.1.5653.1 Galois orbit size: $1$ Smallest permutation container: $S_5$ Parity: even Determinant: 1.5653.2t1.a.a Projective image: $S_5$ Projective stem field: Galois closure of 5.1.5653.1

## Defining polynomial

 $f(x)$ $=$ $$x^{5} - x^{4} + 2x - 1$$ x^5 - x^4 + 2*x - 1 .

The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: $$x^{2} + 21x + 5$$

Roots:
 $r_{ 1 }$ $=$ $$21 a + 20 + \left(10 a + 10\right)\cdot 23 + \left(4 a + 15\right)\cdot 23^{2} + \left(18 a + 17\right)\cdot 23^{3} + 8 a\cdot 23^{4} +O(23^{5})$$ 21*a + 20 + (10*a + 10)*23 + (4*a + 15)*23^2 + (18*a + 17)*23^3 + 8*a*23^4+O(23^5) $r_{ 2 }$ $=$ $$17 + 2\cdot 23 + 20\cdot 23^{2} + 17\cdot 23^{3} + 21\cdot 23^{4} +O(23^{5})$$ 17 + 2*23 + 20*23^2 + 17*23^3 + 21*23^4+O(23^5) $r_{ 3 }$ $=$ $$2 a + 16 + \left(12 a + 11\right)\cdot 23 + \left(18 a + 13\right)\cdot 23^{2} + \left(4 a + 3\right)\cdot 23^{3} + 14 a\cdot 23^{4} +O(23^{5})$$ 2*a + 16 + (12*a + 11)*23 + (18*a + 13)*23^2 + (4*a + 3)*23^3 + 14*a*23^4+O(23^5) $r_{ 4 }$ $=$ $$11 a + 9 + \left(21 a + 17\right)\cdot 23 + \left(5 a + 14\right)\cdot 23^{2} + \left(19 a + 21\right)\cdot 23^{3} + \left(17 a + 14\right)\cdot 23^{4} +O(23^{5})$$ 11*a + 9 + (21*a + 17)*23 + (5*a + 14)*23^2 + (19*a + 21)*23^3 + (17*a + 14)*23^4+O(23^5) $r_{ 5 }$ $=$ $$12 a + 8 + \left(a + 3\right)\cdot 23 + \left(17 a + 5\right)\cdot 23^{2} + \left(3 a + 8\right)\cdot 23^{3} + \left(5 a + 8\right)\cdot 23^{4} +O(23^{5})$$ 12*a + 8 + (a + 3)*23 + (17*a + 5)*23^2 + (3*a + 8)*23^3 + (5*a + 8)*23^4+O(23^5)

## 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 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.