# Properties

 Label 2.2352.6t3.d Dimension $2$ Group $D_{6}$ Conductor $2352$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{6}$ Conductor: $$2352$$$$\medspace = 2^{4} \cdot 3 \cdot 7^{2}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 6.2.38723328.1 Galois orbit size: $1$ Smallest permutation container: $D_{6}$ Parity: odd Projective image: $S_3$ Projective field: Galois closure of 3.1.588.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $$x^{2} + 24x + 2$$
Roots:
 $r_{ 1 }$ $=$ $$6 a + 20 + \left(19 a + 12\right)\cdot 29 + \left(6 a + 27\right)\cdot 29^{2} + \left(14 a + 13\right)\cdot 29^{3} + 7\cdot 29^{4} + \left(27 a + 6\right)\cdot 29^{5} +O(29^{6})$$ 6*a + 20 + (19*a + 12)*29 + (6*a + 27)*29^2 + (14*a + 13)*29^3 + 7*29^4 + (27*a + 6)*29^5+O(29^6) $r_{ 2 }$ $=$ $$12 + 28\cdot 29 + 10\cdot 29^{2} + 5\cdot 29^{3} + 3\cdot 29^{4} + 2\cdot 29^{5} +O(29^{6})$$ 12 + 28*29 + 10*29^2 + 5*29^3 + 3*29^4 + 2*29^5+O(29^6) $r_{ 3 }$ $=$ $$23 a + 21 + \left(9 a + 15\right)\cdot 29 + \left(22 a + 12\right)\cdot 29^{2} + \left(14 a + 20\right)\cdot 29^{3} + \left(28 a + 24\right)\cdot 29^{4} + \left(a + 24\right)\cdot 29^{5} +O(29^{6})$$ 23*a + 21 + (9*a + 15)*29 + (22*a + 12)*29^2 + (14*a + 20)*29^3 + (28*a + 24)*29^4 + (a + 24)*29^5+O(29^6) $r_{ 4 }$ $=$ $$23 a + 9 + \left(9 a + 16\right)\cdot 29 + \left(22 a + 1\right)\cdot 29^{2} + \left(14 a + 15\right)\cdot 29^{3} + \left(28 a + 21\right)\cdot 29^{4} + \left(a + 22\right)\cdot 29^{5} +O(29^{6})$$ 23*a + 9 + (9*a + 16)*29 + (22*a + 1)*29^2 + (14*a + 15)*29^3 + (28*a + 21)*29^4 + (a + 22)*29^5+O(29^6) $r_{ 5 }$ $=$ $$17 + 18\cdot 29^{2} + 23\cdot 29^{3} + 25\cdot 29^{4} + 26\cdot 29^{5} +O(29^{6})$$ 17 + 18*29^2 + 23*29^3 + 25*29^4 + 26*29^5+O(29^6) $r_{ 6 }$ $=$ $$6 a + 8 + \left(19 a + 13\right)\cdot 29 + \left(6 a + 16\right)\cdot 29^{2} + \left(14 a + 8\right)\cdot 29^{3} + 4\cdot 29^{4} + \left(27 a + 4\right)\cdot 29^{5} +O(29^{6})$$ 6*a + 8 + (19*a + 13)*29 + (6*a + 16)*29^2 + (14*a + 8)*29^3 + 4*29^4 + (27*a + 4)*29^5+O(29^6)

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

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

### Character values on conjugacy classes

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