# Properties

 Label 2.2008.7t2.a Dimension 2 Group $D_{7}$ Conductor $2^{3} \cdot 251$ Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{7}$ Conductor: $2008= 2^{3} \cdot 251$ Artin number field: Splitting field of $f= x^{7} - 3 x^{6} + 6 x^{5} - 14 x^{4} + 15 x^{3} - x^{2} + 82 x - 134$ over $\Q$ Size of Galois orbit: 3 Smallest containing permutation representation: $D_{7}$ Parity: Odd Projective image: $D_7$ Projective field: Galois closure of 7.1.8096384512.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: $x^{2} + 45 x + 5$
Roots:
 $r_{ 1 }$ $=$ $17 a + 15 + \left(41 a + 5\right)\cdot 47 + \left(16 a + 23\right)\cdot 47^{2} + \left(22 a + 6\right)\cdot 47^{3} + \left(41 a + 46\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ $r_{ 2 }$ $=$ $46 a + 22 + \left(12 a + 26\right)\cdot 47 + \left(18 a + 26\right)\cdot 47^{2} + \left(29 a + 44\right)\cdot 47^{3} + \left(38 a + 31\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ $r_{ 3 }$ $=$ $30 a + 2 + \left(5 a + 24\right)\cdot 47 + \left(30 a + 15\right)\cdot 47^{2} + \left(24 a + 34\right)\cdot 47^{3} + \left(5 a + 12\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ $r_{ 4 }$ $=$ $a + 20 + \left(34 a + 6\right)\cdot 47 + \left(28 a + 3\right)\cdot 47^{2} + \left(17 a + 38\right)\cdot 47^{3} + \left(8 a + 32\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ $r_{ 5 }$ $=$ $2 a + 22 + \left(36 a + 26\right)\cdot 47 + \left(23 a + 15\right)\cdot 47^{2} + \left(4 a + 10\right)\cdot 47^{3} + \left(6 a + 31\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ $r_{ 6 }$ $=$ $37 + 2\cdot 47 + 30\cdot 47^{2} + 11\cdot 47^{3} + 41\cdot 47^{4} +O\left(47^{ 5 }\right)$ $r_{ 7 }$ $=$ $45 a + 26 + \left(10 a + 2\right)\cdot 47 + \left(23 a + 27\right)\cdot 47^{2} + \left(42 a + 42\right)\cdot 47^{3} + \left(40 a + 38\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$

### Generators of the action on the roots $r_1, \ldots, r_{ 7 }$

 Cycle notation $(1,7)(2,3)(4,6)$ $(1,3)(2,4)(5,7)$

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 7 }$ Character values $c1$ $c2$ $c3$ $1$ $1$ $()$ $2$ $2$ $2$ $7$ $2$ $(1,7)(2,3)(4,6)$ $0$ $0$ $0$ $2$ $7$ $(1,5,7,3,4,6,2)$ $-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - 1$ $\zeta_{7}^{4} + \zeta_{7}^{3}$ $\zeta_{7}^{5} + \zeta_{7}^{2}$ $2$ $7$ $(1,7,4,2,5,3,6)$ $\zeta_{7}^{5} + \zeta_{7}^{2}$ $-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - 1$ $\zeta_{7}^{4} + \zeta_{7}^{3}$ $2$ $7$ $(1,3,2,7,6,5,4)$ $\zeta_{7}^{4} + \zeta_{7}^{3}$ $\zeta_{7}^{5} + \zeta_{7}^{2}$ $-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - 1$
The blue line marks the conjugacy class containing complex conjugation.