# Properties

 Label 2.47.5t2.a.b Dimension $2$ Group $D_{5}$ Conductor $47$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{5}$ Conductor: $$47$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 5.1.2209.1 Galois orbit size: $2$ Smallest permutation container: $D_{5}$ Parity: odd Determinant: 1.47.2t1.a.a Projective image: $D_5$ Projective stem field: Galois closure of 5.1.2209.1

## Defining polynomial

 $f(x)$ $=$ $$x^{5} - 2x^{4} + 2x^{3} - x^{2} + 1$$ x^5 - 2*x^4 + 2*x^3 - x^2 + 1 .

The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: $$x^{2} + 7x + 2$$

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

## Generators of the action on the roots $r_1, \ldots, r_{ 5 }$

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

## Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 5 }$ Character value $1$ $1$ $()$ $2$ $5$ $2$ $(1,3)(2,5)$ $0$ $2$ $5$ $(1,5,4,2,3)$ $-\zeta_{5}^{3} - \zeta_{5}^{2} - 1$ $2$ $5$ $(1,4,3,5,2)$ $\zeta_{5}^{3} + \zeta_{5}^{2}$

The blue line marks the conjugacy class containing complex conjugation.