# Properties

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

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## 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 stem field: Galois closure of 6.2.38720000.1 Galois orbit size: $1$ Smallest permutation container: $D_{6}$ Parity: odd Determinant: 1.88.2t1.b.a Projective image: $S_3$ Projective stem field: Galois closure of 3.1.2200.1

## Defining polynomial

 $f(x)$ $=$ $$x^{6} - 14x^{4} - 30x^{3} - 38x^{2} - 60x - 47$$ x^6 - 14*x^4 - 30*x^3 - 38*x^2 - 60*x - 47 .

The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 8.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: $$x^{2} + 16x + 3$$

Roots:
 $r_{ 1 }$ $=$ $$9 + 4\cdot 17 + 10\cdot 17^{2} + 2\cdot 17^{3} + 10\cdot 17^{4} + 9\cdot 17^{5} + 11\cdot 17^{6} + 13\cdot 17^{7} +O(17^{8})$$ 9 + 4*17 + 10*17^2 + 2*17^3 + 10*17^4 + 9*17^5 + 11*17^6 + 13*17^7+O(17^8) $r_{ 2 }$ $=$ $$4 a + 12 + 4\cdot 17 + \left(4 a + 14\right)\cdot 17^{2} + 9 a\cdot 17^{3} + \left(7 a + 7\right)\cdot 17^{4} + \left(7 a + 15\right)\cdot 17^{5} + \left(6 a + 14\right)\cdot 17^{6} + \left(10 a + 5\right)\cdot 17^{7} +O(17^{8})$$ 4*a + 12 + 4*17 + (4*a + 14)*17^2 + 9*a*17^3 + (7*a + 7)*17^4 + (7*a + 15)*17^5 + (6*a + 14)*17^6 + (10*a + 5)*17^7+O(17^8) $r_{ 3 }$ $=$ $$5 a + 7 + \left(a + 9\right)\cdot 17 + \left(3 a + 3\right)\cdot 17^{2} + \left(16 a + 13\right)\cdot 17^{3} + \left(11 a + 2\right)\cdot 17^{4} + \left(6 a + 4\right)\cdot 17^{5} + \left(6 a + 4\right)\cdot 17^{6} + \left(15 a + 15\right)\cdot 17^{7} +O(17^{8})$$ 5*a + 7 + (a + 9)*17 + (3*a + 3)*17^2 + (16*a + 13)*17^3 + (11*a + 2)*17^4 + (6*a + 4)*17^5 + (6*a + 4)*17^6 + (15*a + 15)*17^7+O(17^8) $r_{ 4 }$ $=$ $$13 a + 16 + 16 a\cdot 17 + \left(12 a + 1\right)\cdot 17^{2} + \left(7 a + 6\right)\cdot 17^{3} + \left(9 a + 5\right)\cdot 17^{4} + \left(9 a + 15\right)\cdot 17^{5} + \left(10 a + 13\right)\cdot 17^{6} + \left(6 a + 9\right)\cdot 17^{7} +O(17^{8})$$ 13*a + 16 + 16*a*17 + (12*a + 1)*17^2 + (7*a + 6)*17^3 + (9*a + 5)*17^4 + (9*a + 15)*17^5 + (10*a + 13)*17^6 + (6*a + 9)*17^7+O(17^8) $r_{ 5 }$ $=$ $$12 a + 12 + \left(15 a + 5\right)\cdot 17 + \left(13 a + 5\right)\cdot 17^{2} + 9\cdot 17^{3} + \left(5 a + 15\right)\cdot 17^{4} + \left(10 a + 15\right)\cdot 17^{5} + \left(10 a + 3\right)\cdot 17^{6} + \left(a + 7\right)\cdot 17^{7} +O(17^{8})$$ 12*a + 12 + (15*a + 5)*17 + (13*a + 5)*17^2 + 9*17^3 + (5*a + 15)*17^4 + (10*a + 15)*17^5 + (10*a + 3)*17^6 + (a + 7)*17^7+O(17^8) $r_{ 6 }$ $=$ $$12 + 8\cdot 17 + 16\cdot 17^{2} + 17^{3} + 10\cdot 17^{4} + 7\cdot 17^{5} + 2\cdot 17^{6} + 16\cdot 17^{7} +O(17^{8})$$ 12 + 8*17 + 16*17^2 + 17^3 + 10*17^4 + 7*17^5 + 2*17^6 + 16*17^7+O(17^8)

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

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

## Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 6 }$ Character value $1$ $1$ $()$ $2$ $1$ $2$ $(1,6)(2,5)(3,4)$ $-2$ $3$ $2$ $(1,2)(3,4)(5,6)$ $0$ $3$ $2$ $(1,3)(4,6)$ $0$ $2$ $3$ $(1,5,3)(2,4,6)$ $-1$ $2$ $6$ $(1,4,5,6,3,2)$ $1$

The blue line marks the conjugacy class containing complex conjugation.