# Properties

 Label 2.2200.6t3.a Dimension $2$ Group $D_{6}$ Conductor $2200$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{6}$ Conductor: $$2200$$$$\medspace = 2^{3} \cdot 5^{2} \cdot 11$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 6.2.193600000.1 Galois orbit size: $1$ Smallest permutation container: $D_{6}$ Parity: odd Projective image: $S_3$ Projective field: Galois closure of 3.1.2200.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: $$x^{2} + 33x + 2$$
Roots:
 $r_{ 1 }$ $=$ $$31 + 22\cdot 37 + 21\cdot 37^{2} + 25\cdot 37^{3} + 3\cdot 37^{4} + 10\cdot 37^{5} +O(37^{6})$$ 31 + 22*37 + 21*37^2 + 25*37^3 + 3*37^4 + 10*37^5+O(37^6) $r_{ 2 }$ $=$ $$36 a + 29 + \left(23 a + 32\right)\cdot 37 + \left(21 a + 5\right)\cdot 37^{2} + \left(17 a + 6\right)\cdot 37^{3} + \left(34 a + 7\right)\cdot 37^{4} + \left(14 a + 6\right)\cdot 37^{5} +O(37^{6})$$ 36*a + 29 + (23*a + 32)*37 + (21*a + 5)*37^2 + (17*a + 6)*37^3 + (34*a + 7)*37^4 + (14*a + 6)*37^5+O(37^6) $r_{ 3 }$ $=$ $$16 a + 8 + \left(31 a + 26\right)\cdot 37 + \left(24 a + 10\right)\cdot 37^{2} + \left(34 a + 4\right)\cdot 37^{3} + \left(33 a + 3\right)\cdot 37^{4} + \left(a + 8\right)\cdot 37^{5} +O(37^{6})$$ 16*a + 8 + (31*a + 26)*37 + (24*a + 10)*37^2 + (34*a + 4)*37^3 + (33*a + 3)*37^4 + (a + 8)*37^5+O(37^6) $r_{ 4 }$ $=$ $$a + 25 + \left(13 a + 18\right)\cdot 37 + \left(15 a + 31\right)\cdot 37^{2} + \left(19 a + 17\right)\cdot 37^{3} + \left(2 a + 16\right)\cdot 37^{4} + \left(22 a + 31\right)\cdot 37^{5} +O(37^{6})$$ a + 25 + (13*a + 18)*37 + (15*a + 31)*37^2 + (19*a + 17)*37^3 + (2*a + 16)*37^4 + (22*a + 31)*37^5+O(37^6) $r_{ 5 }$ $=$ $$20 + 22\cdot 37 + 36\cdot 37^{2} + 12\cdot 37^{3} + 13\cdot 37^{4} + 36\cdot 37^{5} +O(37^{6})$$ 20 + 22*37 + 36*37^2 + 12*37^3 + 13*37^4 + 36*37^5+O(37^6) $r_{ 6 }$ $=$ $$21 a + 35 + \left(5 a + 24\right)\cdot 37 + \left(12 a + 4\right)\cdot 37^{2} + \left(2 a + 7\right)\cdot 37^{3} + \left(3 a + 30\right)\cdot 37^{4} + \left(35 a + 18\right)\cdot 37^{5} +O(37^{6})$$ 21*a + 35 + (5*a + 24)*37 + (12*a + 4)*37^2 + (2*a + 7)*37^3 + (3*a + 30)*37^4 + (35*a + 18)*37^5+O(37^6)

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

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

### Character values on conjugacy classes

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