# Properties

 Label 2.119.5t2.a Dimension $2$ Group $D_{5}$ Conductor $119$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{5}$ Conductor: $$119$$$$\medspace = 7 \cdot 17$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 5.1.14161.1 Galois orbit size: $2$ Smallest permutation container: $D_{5}$ Parity: odd Projective image: $D_5$ Projective field: 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:
 $r_{ 1 }$ $=$ $$5 a + 3 + \left(3 a + 10\right)\cdot 11 + \left(5 a + 6\right)\cdot 11^{2} + \left(9 a + 4\right)\cdot 11^{3} + \left(10 a + 8\right)\cdot 11^{4} +O(11^{5})$$ $r_{ 2 }$ $=$ $$6 a + 1 + \left(7 a + 8\right)\cdot 11 + \left(5 a + 2\right)\cdot 11^{2} + \left(a + 4\right)\cdot 11^{3} + 9\cdot 11^{4} +O(11^{5})$$ $r_{ 3 }$ $=$ $$7 + 5\cdot 11 + 2\cdot 11^{2} + 6\cdot 11^{3} + 7\cdot 11^{4} +O(11^{5})$$ $r_{ 4 }$ $=$ $$3 a + \left(9 a + 4\right)\cdot 11 + \left(6 a + 1\right)\cdot 11^{2} + \left(8 a + 10\right)\cdot 11^{4} +O(11^{5})$$ $r_{ 5 }$ $=$ $$8 a + 1 + \left(a + 5\right)\cdot 11 + \left(4 a + 8\right)\cdot 11^{2} + \left(10 a + 6\right)\cdot 11^{3} + \left(2 a + 8\right)\cdot 11^{4} +O(11^{5})$$

### 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 values $c1$ $c2$ $1$ $1$ $()$ $2$ $2$ $5$ $2$ $(1,2)(4,5)$ $0$ $0$ $2$ $5$ $(1,3,2,5,4)$ $\zeta_{5}^{3} + \zeta_{5}^{2}$ $-\zeta_{5}^{3} - \zeta_{5}^{2} - 1$ $2$ $5$ $(1,2,4,3,5)$ $-\zeta_{5}^{3} - \zeta_{5}^{2} - 1$ $\zeta_{5}^{3} + \zeta_{5}^{2}$
The blue line marks the conjugacy class containing complex conjugation.