# Properties

 Label 2.127.5t2.a Dimension $2$ Group $D_{5}$ Conductor $127$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{5}$ Conductor: $$127$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 5.1.16129.1 Galois orbit size: $2$ Smallest permutation container: $D_{5}$ Parity: odd Projective image: $D_5$ Projective field: Galois closure of 5.1.16129.1

## Galois action

### Roots of defining polynomial

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

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

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

### Character values on conjugacy classes

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