# Properties

 Label 2.2200.6t3.d 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 6.2.24200000.1 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 $\Q_{ 41 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: $$x^{2} + 38x + 6$$
Roots:
 $r_{ 1 }$ $=$ $$10 + 36\cdot 41 + 29\cdot 41^{2} + 10\cdot 41^{3} + 5\cdot 41^{4} + 33\cdot 41^{5} +O(41^{6})$$ 10 + 36*41 + 29*41^2 + 10*41^3 + 5*41^4 + 33*41^5+O(41^6) $r_{ 2 }$ $=$ $$32 + 4\cdot 41 + 11\cdot 41^{2} + 30\cdot 41^{3} + 35\cdot 41^{4} + 7\cdot 41^{5} +O(41^{6})$$ 32 + 4*41 + 11*41^2 + 30*41^3 + 35*41^4 + 7*41^5+O(41^6) $r_{ 3 }$ $=$ $$2 a + 10 + \left(20 a + 3\right)\cdot 41 + \left(4 a + 40\right)\cdot 41^{2} + \left(35 a + 20\right)\cdot 41^{3} + \left(32 a + 33\right)\cdot 41^{4} + 7\cdot 41^{5} +O(41^{6})$$ 2*a + 10 + (20*a + 3)*41 + (4*a + 40)*41^2 + (35*a + 20)*41^3 + (32*a + 33)*41^4 + 7*41^5+O(41^6) $r_{ 4 }$ $=$ $$39 a + 32 + \left(20 a + 37\right)\cdot 41 + 36 a\cdot 41^{2} + \left(5 a + 20\right)\cdot 41^{3} + \left(8 a + 7\right)\cdot 41^{4} + \left(40 a + 33\right)\cdot 41^{5} +O(41^{6})$$ 39*a + 32 + (20*a + 37)*41 + 36*a*41^2 + (5*a + 20)*41^3 + (8*a + 7)*41^4 + (40*a + 33)*41^5+O(41^6) $r_{ 5 }$ $=$ $$39 a + 16 + \left(20 a + 20\right)\cdot 41 + \left(36 a + 33\right)\cdot 41^{2} + \left(5 a + 39\right)\cdot 41^{3} + \left(8 a + 14\right)\cdot 41^{4} + \left(40 a + 18\right)\cdot 41^{5} +O(41^{6})$$ 39*a + 16 + (20*a + 20)*41 + (36*a + 33)*41^2 + (5*a + 39)*41^3 + (8*a + 14)*41^4 + (40*a + 18)*41^5+O(41^6) $r_{ 6 }$ $=$ $$2 a + 26 + \left(20 a + 20\right)\cdot 41 + \left(4 a + 7\right)\cdot 41^{2} + \left(35 a + 1\right)\cdot 41^{3} + \left(32 a + 26\right)\cdot 41^{4} + 22\cdot 41^{5} +O(41^{6})$$ 2*a + 26 + (20*a + 20)*41 + (4*a + 7)*41^2 + (35*a + 1)*41^3 + (32*a + 26)*41^4 + 22*41^5+O(41^6)

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

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

### Character values on conjugacy classes

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