# Properties

 Label 3.3627.6t11.b.a Dimension $3$ Group $S_4\times C_2$ Conductor $3627$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $3$ Group: $S_4\times C_2$ Conductor: $$3627$$$$\medspace = 3^{2} \cdot 13 \cdot 31$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 6.0.3485547.1 Galois orbit size: $1$ Smallest permutation container: $S_4\times C_2$ Parity: odd Determinant: 1.403.2t1.a.a Projective image: $S_4$ Projective stem field: Galois closure of 4.2.47151.1

## Defining polynomial

 $f(x)$ $=$ $$x^{6} - 2x^{5} - 5x^{4} - 2x^{3} + 17x^{2} + 55x + 47$$ x^6 - 2*x^5 - 5*x^4 - 2*x^3 + 17*x^2 + 55*x + 47 .

The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: $$x^{2} + 33x + 2$$

Roots:
 $r_{ 1 }$ $=$ $$1 + 15\cdot 37 + 19\cdot 37^{2} + 37^{3} + 6\cdot 37^{4} + 3\cdot 37^{5} + 6\cdot 37^{6} + 10\cdot 37^{7} + 20\cdot 37^{8} + 20\cdot 37^{9} +O(37^{10})$$ 1 + 15*37 + 19*37^2 + 37^3 + 6*37^4 + 3*37^5 + 6*37^6 + 10*37^7 + 20*37^8 + 20*37^9+O(37^10) $r_{ 2 }$ $=$ $$26 a + 32 + \left(36 a + 3\right)\cdot 37 + \left(8 a + 16\right)\cdot 37^{2} + \left(a + 28\right)\cdot 37^{3} + \left(20 a + 33\right)\cdot 37^{4} + \left(7 a + 5\right)\cdot 37^{5} + \left(6 a + 25\right)\cdot 37^{6} + \left(36 a + 2\right)\cdot 37^{7} + \left(33 a + 20\right)\cdot 37^{8} + \left(19 a + 12\right)\cdot 37^{9} +O(37^{10})$$ 26*a + 32 + (36*a + 3)*37 + (8*a + 16)*37^2 + (a + 28)*37^3 + (20*a + 33)*37^4 + (7*a + 5)*37^5 + (6*a + 25)*37^6 + (36*a + 2)*37^7 + (33*a + 20)*37^8 + (19*a + 12)*37^9+O(37^10) $r_{ 3 }$ $=$ $$15 a + 2 + \left(8 a + 29\right)\cdot 37 + \left(35 a + 22\right)\cdot 37^{2} + \left(22 a + 30\right)\cdot 37^{3} + \left(25 a + 7\right)\cdot 37^{4} + \left(16 a + 20\right)\cdot 37^{5} + \left(14 a + 28\right)\cdot 37^{6} + 5 a\cdot 37^{7} + \left(6 a + 25\right)\cdot 37^{8} + \left(26 a + 21\right)\cdot 37^{9} +O(37^{10})$$ 15*a + 2 + (8*a + 29)*37 + (35*a + 22)*37^2 + (22*a + 30)*37^3 + (25*a + 7)*37^4 + (16*a + 20)*37^5 + (14*a + 28)*37^6 + 5*a*37^7 + (6*a + 25)*37^8 + (26*a + 21)*37^9+O(37^10) $r_{ 4 }$ $=$ $$22 a + 25 + \left(28 a + 10\right)\cdot 37 + \left(a + 7\right)\cdot 37^{2} + \left(14 a + 13\right)\cdot 37^{3} + \left(11 a + 13\right)\cdot 37^{4} + \left(20 a + 24\right)\cdot 37^{5} + \left(22 a + 32\right)\cdot 37^{6} + \left(31 a + 7\right)\cdot 37^{7} + \left(30 a + 7\right)\cdot 37^{8} + \left(10 a + 9\right)\cdot 37^{9} +O(37^{10})$$ 22*a + 25 + (28*a + 10)*37 + (a + 7)*37^2 + (14*a + 13)*37^3 + (11*a + 13)*37^4 + (20*a + 24)*37^5 + (22*a + 32)*37^6 + (31*a + 7)*37^7 + (30*a + 7)*37^8 + (10*a + 9)*37^9+O(37^10) $r_{ 5 }$ $=$ $$11 a + 25 + 13\cdot 37 + \left(28 a + 15\right)\cdot 37^{2} + \left(35 a + 24\right)\cdot 37^{3} + \left(16 a + 1\right)\cdot 37^{4} + \left(29 a + 16\right)\cdot 37^{5} + \left(30 a + 5\right)\cdot 37^{6} + 30\cdot 37^{7} + \left(3 a + 8\right)\cdot 37^{8} + \left(17 a + 21\right)\cdot 37^{9} +O(37^{10})$$ 11*a + 25 + 13*37 + (28*a + 15)*37^2 + (35*a + 24)*37^3 + (16*a + 1)*37^4 + (29*a + 16)*37^5 + (30*a + 5)*37^6 + 30*37^7 + (3*a + 8)*37^8 + (17*a + 21)*37^9+O(37^10) $r_{ 6 }$ $=$ $$28 + 37 + 30\cdot 37^{2} + 12\cdot 37^{3} + 11\cdot 37^{4} + 4\cdot 37^{5} + 13\cdot 37^{6} + 22\cdot 37^{7} + 29\cdot 37^{8} + 25\cdot 37^{9} +O(37^{10})$$ 28 + 37 + 30*37^2 + 12*37^3 + 11*37^4 + 4*37^5 + 13*37^6 + 22*37^7 + 29*37^8 + 25*37^9+O(37^10)

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

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

## Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 6 }$ Character value $1$ $1$ $()$ $3$ $1$ $2$ $(1,6)(2,4)(3,5)$ $-3$ $3$ $2$ $(2,4)(3,5)$ $-1$ $3$ $2$ $(3,5)$ $1$ $6$ $2$ $(2,3)(4,5)$ $-1$ $6$ $2$ $(1,2)(3,5)(4,6)$ $1$ $8$ $3$ $(1,2,3)(4,5,6)$ $0$ $6$ $4$ $(2,3,4,5)$ $-1$ $6$ $4$ $(1,6)(2,3,4,5)$ $1$ $8$ $6$ $(1,3,4,6,5,2)$ $0$

The blue line marks the conjugacy class containing complex conjugation.