# Properties

 Label 2.131.5t2.a Dimension 2 Group $D_{5}$ Conductor $131$ Frobenius-Schur indicator 1

# Learn more about

## Basic invariants

 Dimension: $2$ Group: $D_{5}$ Conductor: $131$ Artin number field: Splitting field of $f= x^{5} - x^{4} + 2 x^{3} - x^{2} + x + 2$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $D_{5}$ Parity: Odd Projective image: $D_5$ Projective field: Galois closure of 5.1.17161.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: $x^{2} + 16 x + 3$
Roots:
 $r_{ 1 }$ $=$ $15 + 10\cdot 17 + 15\cdot 17^{2} + 13\cdot 17^{3} + 4\cdot 17^{4} +O\left(17^{ 5 }\right)$ $r_{ 2 }$ $=$ $12 a + 1 + \left(9 a + 12\right)\cdot 17 + \left(15 a + 2\right)\cdot 17^{2} + \left(15 a + 11\right)\cdot 17^{3} + \left(14 a + 1\right)\cdot 17^{4} +O\left(17^{ 5 }\right)$ $r_{ 3 }$ $=$ $5 a + 13 + \left(7 a + 9\right)\cdot 17 + \left(a + 8\right)\cdot 17^{2} + \left(a + 11\right)\cdot 17^{3} + 2 a\cdot 17^{4} +O\left(17^{ 5 }\right)$ $r_{ 4 }$ $=$ $6 a + \left(12 a + 6\right)\cdot 17 + a\cdot 17^{2} + \left(13 a + 10\right)\cdot 17^{3} + \left(14 a + 12\right)\cdot 17^{4} +O\left(17^{ 5 }\right)$ $r_{ 5 }$ $=$ $11 a + 6 + \left(4 a + 12\right)\cdot 17 + \left(15 a + 6\right)\cdot 17^{2} + \left(3 a + 4\right)\cdot 17^{3} + \left(2 a + 14\right)\cdot 17^{4} +O\left(17^{ 5 }\right)$

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

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

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 5 }$ Character values $c1$ $c2$ $1$ $1$ $()$ $2$ $2$ $5$ $2$ $(1,2)(3,4)$ $0$ $0$ $2$ $5$ $(1,4,3,2,5)$ $\zeta_{5}^{3} + \zeta_{5}^{2}$ $-\zeta_{5}^{3} - \zeta_{5}^{2} - 1$ $2$ $5$ $(1,3,5,4,2)$ $-\zeta_{5}^{3} - \zeta_{5}^{2} - 1$ $\zeta_{5}^{3} + \zeta_{5}^{2}$
The blue line marks the conjugacy class containing complex conjugation.