# Properties

 Label 2.119.5t2.a.b Dimension 2 Group $D_{5}$ Conductor $7 \cdot 17$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{5}$ Conductor: $119= 7 \cdot 17$ Artin number field: Splitting field of 5.1.14161.1 defined by $f= x^{5} - x^{4} - x^{2} + 3 x - 1$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $D_{5}$ Parity: Odd Determinant: 1.119.2t1.a.a Projective image: $D_5$ Projective field: Galois closure of 5.1.14161.1

## Galois action

### Roots of defining polynomial

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

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

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

### Character values on conjugacy classes

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