# Properties

 Label 2.648.6t3.b Dimension $2$ Group $D_{6}$ Conductor $648$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{6}$ Conductor: $$648$$$$\medspace = 2^{3} \cdot 3^{4}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 6.0.1259712.1 Galois orbit size: $1$ Smallest permutation container: $D_{6}$ Parity: odd Projective image: $S_3$ Projective field: 3.1.648.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 7 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 7 }$: $$x^{2} + 6 x + 3$$
Roots:
 $r_{ 1 }$ $=$ $$3 + 2\cdot 7 + 6\cdot 7^{2} + 6\cdot 7^{3} + 6\cdot 7^{4} + 6\cdot 7^{5} + 6\cdot 7^{6} + 7^{7} +O(7^{8})$$ $r_{ 2 }$ $=$ $$2 + 5\cdot 7 + 4\cdot 7^{2} + 6\cdot 7^{3} + 2\cdot 7^{4} + 3\cdot 7^{5} + 7^{6} + 4\cdot 7^{7} +O(7^{8})$$ $r_{ 3 }$ $=$ $$2 a + 1 + 3\cdot 7 + \left(2 a + 6\right)\cdot 7^{2} + \left(5 a + 1\right)\cdot 7^{4} + \left(5 a + 3\right)\cdot 7^{5} + \left(3 a + 4\right)\cdot 7^{6} +O(7^{8})$$ $r_{ 4 }$ $=$ $$6 a + 3 + \left(a + 6\right)\cdot 7 + \left(5 a + 2\right)\cdot 7^{2} + \left(2 a + 1\right)\cdot 7^{3} + \left(a + 6\right)\cdot 7^{4} + 5\cdot 7^{5} + \left(a + 5\right)\cdot 7^{6} + \left(6 a + 5\right)\cdot 7^{7} +O(7^{8})$$ $r_{ 5 }$ $=$ $$5 a + 3 + \left(6 a + 1\right)\cdot 7 + \left(4 a + 1\right)\cdot 7^{2} + \left(6 a + 6\right)\cdot 7^{3} + \left(a + 5\right)\cdot 7^{4} + \left(a + 3\right)\cdot 7^{5} + \left(3 a + 2\right)\cdot 7^{6} + \left(6 a + 4\right)\cdot 7^{7} +O(7^{8})$$ $r_{ 6 }$ $=$ $$a + 2 + \left(5 a + 2\right)\cdot 7 + \left(a + 6\right)\cdot 7^{2} + \left(4 a + 5\right)\cdot 7^{3} + \left(5 a + 4\right)\cdot 7^{4} + \left(6 a + 4\right)\cdot 7^{5} + \left(5 a + 6\right)\cdot 7^{6} + 3\cdot 7^{7} +O(7^{8})$$

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

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

### Character values on conjugacy classes

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