# Properties

 Label 2.1216.6t3.d Dimension $2$ Group $D_{6}$ Conductor $1216$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{6}$ Conductor: $$1216$$$$\medspace = 2^{6} \cdot 19$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 6.0.739328.2 Galois orbit size: $1$ Smallest permutation container: $D_{6}$ Parity: odd Projective image: $S_3$ Projective field: Galois closure of 3.1.76.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: $$x^{2} + 38x + 6$$
Roots:
 $r_{ 1 }$ $=$ $$4 + 10\cdot 41 + 2\cdot 41^{2} + 28\cdot 41^{3} +O(41^{5})$$ 4 + 10*41 + 2*41^2 + 28*41^3+O(41^5) $r_{ 2 }$ $=$ $$22 a + 32 + \left(38 a + 8\right)\cdot 41 + \left(9 a + 2\right)\cdot 41^{2} + \left(7 a + 5\right)\cdot 41^{3} + \left(19 a + 4\right)\cdot 41^{4} +O(41^{5})$$ 22*a + 32 + (38*a + 8)*41 + (9*a + 2)*41^2 + (7*a + 5)*41^3 + (19*a + 4)*41^4+O(41^5) $r_{ 3 }$ $=$ $$19 a + 16 + \left(2 a + 20\right)\cdot 41 + \left(31 a + 34\right)\cdot 41^{2} + \left(33 a + 16\right)\cdot 41^{3} + \left(21 a + 13\right)\cdot 41^{4} +O(41^{5})$$ 19*a + 16 + (2*a + 20)*41 + (31*a + 34)*41^2 + (33*a + 16)*41^3 + (21*a + 13)*41^4+O(41^5) $r_{ 4 }$ $=$ $$37 + 30\cdot 41 + 38\cdot 41^{2} + 12\cdot 41^{3} + 40\cdot 41^{4} +O(41^{5})$$ 37 + 30*41 + 38*41^2 + 12*41^3 + 40*41^4+O(41^5) $r_{ 5 }$ $=$ $$19 a + 9 + \left(2 a + 32\right)\cdot 41 + \left(31 a + 38\right)\cdot 41^{2} + \left(33 a + 35\right)\cdot 41^{3} + \left(21 a + 36\right)\cdot 41^{4} +O(41^{5})$$ 19*a + 9 + (2*a + 32)*41 + (31*a + 38)*41^2 + (33*a + 35)*41^3 + (21*a + 36)*41^4+O(41^5) $r_{ 6 }$ $=$ $$22 a + 25 + \left(38 a + 20\right)\cdot 41 + \left(9 a + 6\right)\cdot 41^{2} + \left(7 a + 24\right)\cdot 41^{3} + \left(19 a + 27\right)\cdot 41^{4} +O(41^{5})$$ 22*a + 25 + (38*a + 20)*41 + (9*a + 6)*41^2 + (7*a + 24)*41^3 + (19*a + 27)*41^4+O(41^5)

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

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

### Character values on conjugacy classes

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