# Properties

 Label 2.3744.6t3.d.a Dimension $2$ Group $D_{6}$ Conductor $3744$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{6}$ Conductor: $$3744$$$$\medspace = 2^{5} \cdot 3^{2} \cdot 13$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 6.0.9345024.1 Galois orbit size: $1$ Smallest permutation container: $D_{6}$ Parity: odd Determinant: 1.104.2t1.b.a Projective image: $S_3$ Projective stem field: Galois closure of 3.1.104.1

## Defining polynomial

 $f(x)$ $=$ $$x^{6} - 2x^{5} + 7x^{4} - 16x^{3} + 25x^{2} - 42x + 31$$ x^6 - 2*x^5 + 7*x^4 - 16*x^3 + 25*x^2 - 42*x + 31 .

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:
 $r_{ 1 }$ $=$ $$24 a + 19 + \left(20 a + 6\right)\cdot 29 + \left(20 a + 16\right)\cdot 29^{2} + \left(7 a + 22\right)\cdot 29^{3} + \left(10 a + 23\right)\cdot 29^{4} + \left(27 a + 17\right)\cdot 29^{5} +O(29^{6})$$ 24*a + 19 + (20*a + 6)*29 + (20*a + 16)*29^2 + (7*a + 22)*29^3 + (10*a + 23)*29^4 + (27*a + 17)*29^5+O(29^6) $r_{ 2 }$ $=$ $$24 + 2\cdot 29 + 8\cdot 29^{2} + 27\cdot 29^{3} + 2\cdot 29^{4} + 6\cdot 29^{5} +O(29^{6})$$ 24 + 2*29 + 8*29^2 + 27*29^3 + 2*29^4 + 6*29^5+O(29^6) $r_{ 3 }$ $=$ $$18 a + 16 + \left(7 a + 17\right)\cdot 29 + \left(27 a + 18\right)\cdot 29^{2} + \left(19 a + 22\right)\cdot 29^{3} + \left(15 a + 12\right)\cdot 29^{4} + \left(28 a + 20\right)\cdot 29^{5} +O(29^{6})$$ 18*a + 16 + (7*a + 17)*29 + (27*a + 18)*29^2 + (19*a + 22)*29^3 + (15*a + 12)*29^4 + (28*a + 20)*29^5+O(29^6) $r_{ 4 }$ $=$ $$5 a + 23 + \left(8 a + 28\right)\cdot 29 + \left(8 a + 11\right)\cdot 29^{2} + \left(21 a + 11\right)\cdot 29^{3} + \left(18 a + 9\right)\cdot 29^{4} + \left(a + 28\right)\cdot 29^{5} +O(29^{6})$$ 5*a + 23 + (8*a + 28)*29 + (8*a + 11)*29^2 + (21*a + 11)*29^3 + (18*a + 9)*29^4 + (a + 28)*29^5+O(29^6) $r_{ 5 }$ $=$ $$11 a + 19 + \left(21 a + 8\right)\cdot 29 + \left(a + 2\right)\cdot 29^{2} + \left(9 a + 8\right)\cdot 29^{3} + \left(13 a + 13\right)\cdot 29^{4} + 2\cdot 29^{5} +O(29^{6})$$ 11*a + 19 + (21*a + 8)*29 + (a + 2)*29^2 + (9*a + 8)*29^3 + (13*a + 13)*29^4 + 2*29^5+O(29^6) $r_{ 6 }$ $=$ $$17 + 22\cdot 29 + 24\cdot 29^{3} + 24\cdot 29^{4} + 11\cdot 29^{5} +O(29^{6})$$ 17 + 22*29 + 24*29^3 + 24*29^4 + 11*29^5+O(29^6)

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.