# Properties

 Label 2.135.6t3.b.a Dimension 2 Group $D_{6}$ Conductor $3^{3} \cdot 5$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{6}$ Conductor: $135= 3^{3} \cdot 5$ Artin number field: Splitting field of $f= x^{6} - 3 x^{5} + 3 x^{4} - 3 x^{2} + 3$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $D_{6}$ Parity: Odd Determinant: 1.15.2t1.a.a

## Galois action

### Roots of defining polynomial

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

### Generators of the action on the roots $r_1, \ldots, r_{ 6 }$

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

### Character values on conjugacy classes

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