# Properties

 Label 2.3332.6t3.e Dimension $2$ Group $D_{6}$ Conductor $3332$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{6}$ Conductor: $$3332$$$$\medspace = 2^{2} \cdot 7^{2} \cdot 17$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 6.0.44408896.1 Galois orbit size: $1$ Smallest permutation container: $D_{6}$ Parity: odd Projective image: $S_3$ Projective field: Galois closure of 3.1.3332.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $$x^{2} + 24x + 2$$
Roots:
 $r_{ 1 }$ $=$ $$14 a + 19 + \left(25 a + 19\right)\cdot 29 + \left(6 a + 3\right)\cdot 29^{2} + \left(2 a + 18\right)\cdot 29^{3} + \left(7 a + 14\right)\cdot 29^{4} + \left(23 a + 15\right)\cdot 29^{5} + \left(16 a + 14\right)\cdot 29^{6} +O(29^{7})$$ 14*a + 19 + (25*a + 19)*29 + (6*a + 3)*29^2 + (2*a + 18)*29^3 + (7*a + 14)*29^4 + (23*a + 15)*29^5 + (16*a + 14)*29^6+O(29^7) $r_{ 2 }$ $=$ $$15 a + 2 + \left(3 a + 17\right)\cdot 29 + \left(22 a + 12\right)\cdot 29^{2} + \left(26 a + 22\right)\cdot 29^{3} + \left(21 a + 18\right)\cdot 29^{4} + \left(5 a + 8\right)\cdot 29^{5} + \left(12 a + 17\right)\cdot 29^{6} +O(29^{7})$$ 15*a + 2 + (3*a + 17)*29 + (22*a + 12)*29^2 + (26*a + 22)*29^3 + (21*a + 18)*29^4 + (5*a + 8)*29^5 + (12*a + 17)*29^6+O(29^7) $r_{ 3 }$ $=$ $$25 + 19\cdot 29 + 16\cdot 29^{3} + 6\cdot 29^{4} + 17\cdot 29^{5} + 10\cdot 29^{6} +O(29^{7})$$ 25 + 19*29 + 16*29^3 + 6*29^4 + 17*29^5 + 10*29^6+O(29^7) $r_{ 4 }$ $=$ $$23 + 9\cdot 29^{2} + 23\cdot 29^{3} + 23\cdot 29^{4} + 19\cdot 29^{5} + 21\cdot 29^{6} +O(29^{7})$$ 23 + 9*29^2 + 23*29^3 + 23*29^4 + 19*29^5 + 21*29^6+O(29^7) $r_{ 5 }$ $=$ $$25 a + 19 + \left(19 a + 6\right)\cdot 29 + \left(4 a + 14\right)\cdot 29^{2} + \left(4 a + 24\right)\cdot 29^{3} + \left(4 a + 17\right)\cdot 29^{4} + \left(25 a + 24\right)\cdot 29^{5} + \left(22 a + 24\right)\cdot 29^{6} +O(29^{7})$$ 25*a + 19 + (19*a + 6)*29 + (4*a + 14)*29^2 + (4*a + 24)*29^3 + (4*a + 17)*29^4 + (25*a + 24)*29^5 + (22*a + 24)*29^6+O(29^7) $r_{ 6 }$ $=$ $$4 a + 28 + \left(9 a + 22\right)\cdot 29 + \left(24 a + 17\right)\cdot 29^{2} + \left(24 a + 11\right)\cdot 29^{3} + \left(24 a + 5\right)\cdot 29^{4} + \left(3 a + 1\right)\cdot 29^{5} + \left(6 a + 27\right)\cdot 29^{6} +O(29^{7})$$ 4*a + 28 + (9*a + 22)*29 + (24*a + 17)*29^2 + (24*a + 11)*29^3 + (24*a + 5)*29^4 + (3*a + 1)*29^5 + (6*a + 27)*29^6+O(29^7)

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

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

### Character values on conjugacy classes

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