# Properties

 Label 2.2175.6t3.d Dimension $2$ Group $D_{6}$ Conductor $2175$ Indicator $1$

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## Basic invariants

 Dimension: $2$ Group: $D_{6}$ Conductor: $$2175$$$$\medspace = 3 \cdot 5^{2} \cdot 29$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 6.2.946125.1 Galois orbit size: $1$ Smallest permutation container: $D_{6}$ Parity: odd Projective image: $S_3$ Projective field: Galois closure of 3.1.87.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: $$x^{2} + 18x + 2$$
Roots:
 $r_{ 1 }$ $=$ $$1 + 10\cdot 19 + 3\cdot 19^{2} + 13\cdot 19^{3} + 10\cdot 19^{4} + 8\cdot 19^{5} + 17\cdot 19^{6} +O(19^{7})$$ 1 + 10*19 + 3*19^2 + 13*19^3 + 10*19^4 + 8*19^5 + 17*19^6+O(19^7) $r_{ 2 }$ $=$ $$11 a + 10 + \left(16 a + 8\right)\cdot 19 + \left(9 a + 6\right)\cdot 19^{2} + 4 a\cdot 19^{3} + \left(9 a + 6\right)\cdot 19^{4} + 3\cdot 19^{5} + \left(14 a + 5\right)\cdot 19^{6} +O(19^{7})$$ 11*a + 10 + (16*a + 8)*19 + (9*a + 6)*19^2 + 4*a*19^3 + (9*a + 6)*19^4 + 3*19^5 + (14*a + 5)*19^6+O(19^7) $r_{ 3 }$ $=$ $$8 a + 2 + \left(2 a + 14\right)\cdot 19 + \left(9 a + 18\right)\cdot 19^{2} + \left(14 a + 13\right)\cdot 19^{3} + \left(9 a + 10\right)\cdot 19^{4} + \left(18 a + 13\right)\cdot 19^{5} + \left(4 a + 18\right)\cdot 19^{6} +O(19^{7})$$ 8*a + 2 + (2*a + 14)*19 + (9*a + 18)*19^2 + (14*a + 13)*19^3 + (9*a + 10)*19^4 + (18*a + 13)*19^5 + (4*a + 18)*19^6+O(19^7) $r_{ 4 }$ $=$ $$10 a + 16 + \left(11 a + 4\right)\cdot 19 + \left(6 a + 15\right)\cdot 19^{2} + \left(13 a + 6\right)\cdot 19^{3} + \left(2 a + 13\right)\cdot 19^{4} + \left(11 a + 7\right)\cdot 19^{5} + \left(8 a + 14\right)\cdot 19^{6} +O(19^{7})$$ 10*a + 16 + (11*a + 4)*19 + (6*a + 15)*19^2 + (13*a + 6)*19^3 + (2*a + 13)*19^4 + (11*a + 7)*19^5 + (8*a + 14)*19^6+O(19^7) $r_{ 5 }$ $=$ $$9 a + 7 + \left(7 a + 6\right)\cdot 19 + \left(12 a + 10\right)\cdot 19^{2} + \left(5 a + 13\right)\cdot 19^{3} + \left(16 a + 2\right)\cdot 19^{4} + \left(7 a + 16\right)\cdot 19^{5} + \left(10 a + 11\right)\cdot 19^{6} +O(19^{7})$$ 9*a + 7 + (7*a + 6)*19 + (12*a + 10)*19^2 + (5*a + 13)*19^3 + (16*a + 2)*19^4 + (7*a + 16)*19^5 + (10*a + 11)*19^6+O(19^7) $r_{ 6 }$ $=$ $$3 + 13\cdot 19 + 2\cdot 19^{2} + 9\cdot 19^{3} + 13\cdot 19^{4} + 7\cdot 19^{5} + 8\cdot 19^{6} +O(19^{7})$$ 3 + 13*19 + 2*19^2 + 9*19^3 + 13*19^4 + 7*19^5 + 8*19^6+O(19^7)

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

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

### Character values on conjugacy classes

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