# Properties

 Label 2.72.6t5.b.b Dimension $2$ Group $S_3\times C_3$ Conductor $72$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $S_3\times C_3$ Conductor: $$72$$$$\medspace = 2^{3} \cdot 3^{2}$$ Artin stem field: Galois closure of 6.0.41472.1 Galois orbit size: $2$ Smallest permutation container: $S_3\times C_3$ Parity: odd Determinant: 1.72.6t1.a.a Projective image: $S_3$ Projective stem field: Galois closure of 3.1.648.1

## Defining polynomial

 $f(x)$ $=$ $$x^{6} - 2x^{5} + 3x^{4} - 2x^{3} + 2x^{2} + 1$$ x^6 - 2*x^5 + 3*x^4 - 2*x^3 + 2*x^2 + 1 .

The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 5.

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

Roots:
 $r_{ 1 }$ $=$ $$24 a + 9 + \left(21 a + 32\right)\cdot 37 + \left(4 a + 18\right)\cdot 37^{2} + \left(31 a + 8\right)\cdot 37^{3} + \left(35 a + 15\right)\cdot 37^{4} +O(37^{5})$$ 24*a + 9 + (21*a + 32)*37 + (4*a + 18)*37^2 + (31*a + 8)*37^3 + (35*a + 15)*37^4+O(37^5) $r_{ 2 }$ $=$ $$13 a + 31 + \left(15 a + 20\right)\cdot 37 + \left(32 a + 15\right)\cdot 37^{2} + \left(5 a + 17\right)\cdot 37^{3} + \left(a + 16\right)\cdot 37^{4} +O(37^{5})$$ 13*a + 31 + (15*a + 20)*37 + (32*a + 15)*37^2 + (5*a + 17)*37^3 + (a + 16)*37^4+O(37^5) $r_{ 3 }$ $=$ $$29 a + 18 + \left(35 a + 13\right)\cdot 37 + \left(27 a + 30\right)\cdot 37^{2} + \left(12 a + 29\right)\cdot 37^{3} + 35\cdot 37^{4} +O(37^{5})$$ 29*a + 18 + (35*a + 13)*37 + (27*a + 30)*37^2 + (12*a + 29)*37^3 + 35*37^4+O(37^5) $r_{ 4 }$ $=$ $$26 a + 1 + \left(16 a + 12\right)\cdot 37 + \left(12 a + 27\right)\cdot 37^{2} + \left(8 a + 27\right)\cdot 37^{3} + \left(24 a + 20\right)\cdot 37^{4} +O(37^{5})$$ 26*a + 1 + (16*a + 12)*37 + (12*a + 27)*37^2 + (8*a + 27)*37^3 + (24*a + 20)*37^4+O(37^5) $r_{ 5 }$ $=$ $$8 a + 23 + \left(a + 16\right)\cdot 37 + \left(9 a + 32\right)\cdot 37^{2} + \left(24 a + 15\right)\cdot 37^{3} + \left(36 a + 24\right)\cdot 37^{4} +O(37^{5})$$ 8*a + 23 + (a + 16)*37 + (9*a + 32)*37^2 + (24*a + 15)*37^3 + (36*a + 24)*37^4+O(37^5) $r_{ 6 }$ $=$ $$11 a + 31 + \left(20 a + 15\right)\cdot 37 + \left(24 a + 23\right)\cdot 37^{2} + \left(28 a + 11\right)\cdot 37^{3} + \left(12 a + 35\right)\cdot 37^{4} +O(37^{5})$$ 11*a + 31 + (20*a + 15)*37 + (24*a + 23)*37^2 + (28*a + 11)*37^3 + (12*a + 35)*37^4+O(37^5)

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.