# Properties

 Label 2.2011.7t2.1c2 Dimension 2 Group $D_{7}$ Conductor $2011$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{7}$ Conductor: $2011$ Artin number field: Splitting field of $f= x^{7} - 2 x^{6} + 13 x^{5} - 13 x^{4} - 29 x^{2} - 30 x - 36$ over $\Q$ Size of Galois orbit: 3 Smallest containing permutation representation: $D_{7}$ Parity: Odd Determinant: 1.2011.2t1.1c1

## 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:
 $r_{ 1 }$ $=$ $4 + 6\cdot 11 + 4\cdot 11^{2} + 7\cdot 11^{3} + 7\cdot 11^{4} +O\left(11^{ 5 }\right)$ $r_{ 2 }$ $=$ $3 a + 8 + \left(7 a + 5\right)\cdot 11 + \left(9 a + 6\right)\cdot 11^{2} + 8\cdot 11^{3} + \left(5 a + 4\right)\cdot 11^{4} +O\left(11^{ 5 }\right)$ $r_{ 3 }$ $=$ $10 a + \left(4 a + 2\right)\cdot 11 + \left(4 a + 2\right)\cdot 11^{2} + 11^{3} + \left(8 a + 4\right)\cdot 11^{4} +O\left(11^{ 5 }\right)$ $r_{ 4 }$ $=$ $7 a + 6 + \left(a + 4\right)\cdot 11 + \left(4 a + 3\right)\cdot 11^{2} + \left(10 a + 5\right)\cdot 11^{3} + 3\cdot 11^{4} +O\left(11^{ 5 }\right)$ $r_{ 5 }$ $=$ $a + 7 + 6 a\cdot 11 + \left(6 a + 4\right)\cdot 11^{2} + \left(10 a + 9\right)\cdot 11^{3} + \left(2 a + 2\right)\cdot 11^{4} +O\left(11^{ 5 }\right)$ $r_{ 6 }$ $=$ $8 a + 9 + \left(3 a + 9\right)\cdot 11 + \left(a + 4\right)\cdot 11^{2} + \left(10 a + 2\right)\cdot 11^{3} + \left(5 a + 2\right)\cdot 11^{4} +O\left(11^{ 5 }\right)$ $r_{ 7 }$ $=$ $4 a + 1 + \left(9 a + 4\right)\cdot 11 + \left(6 a + 7\right)\cdot 11^{2} + 9\cdot 11^{3} + \left(10 a + 7\right)\cdot 11^{4} +O\left(11^{ 5 }\right)$

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

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

### Character values on conjugacy classes

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