# Properties

 Label 2.79.5t2.a Dimension 2 Group $D_{5}$ Conductor $79$ Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{5}$ Conductor: $79$ Artin number field: Splitting field of $f= x^{5} - x^{4} + x^{3} - 2 x^{2} + 3 x - 1$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $D_{5}$ Parity: Odd Projective image: $D_5$ Projective field: Galois closure of 5.1.6241.1

## 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 $\mathbb{Q}_{ 7 }$: $x^{2} + 6 x + 3$
Roots:
 $r_{ 1 }$ $=$ $3 a + 3 + 3 a\cdot 7 + \left(4 a + 2\right)\cdot 7^{2} + \left(4 a + 5\right)\cdot 7^{3} + \left(4 a + 6\right)\cdot 7^{4} +O\left(7^{ 5 }\right)$ $r_{ 2 }$ $=$ $2 + 3\cdot 7 + 4\cdot 7^{2} + 2\cdot 7^{4} +O\left(7^{ 5 }\right)$ $r_{ 3 }$ $=$ $4 a + \left(5 a + 4\right)\cdot 7 + \left(4 a + 2\right)\cdot 7^{2} + 3\cdot 7^{3} + \left(a + 2\right)\cdot 7^{4} +O\left(7^{ 5 }\right)$ $r_{ 4 }$ $=$ $3 a + 4 + \left(a + 5\right)\cdot 7 + \left(2 a + 1\right)\cdot 7^{2} + \left(6 a + 6\right)\cdot 7^{3} + \left(5 a + 2\right)\cdot 7^{4} +O\left(7^{ 5 }\right)$ $r_{ 5 }$ $=$ $4 a + 6 + 3 a\cdot 7 + \left(2 a + 3\right)\cdot 7^{2} + \left(2 a + 5\right)\cdot 7^{3} + \left(2 a + 6\right)\cdot 7^{4} +O\left(7^{ 5 }\right)$

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