Properties

 Label 1.72.6t1.d.b Dimension $1$ Group $C_6$ Conductor $72$ Root number not computed Indicator $0$

Related objects

Basic invariants

 Dimension: $1$ Group: $C_6$ Conductor: $$72$$$$\medspace = 2^{3} \cdot 3^{2}$$ Artin field: Galois closure of 6.6.3359232.1 Galois orbit size: $2$ Smallest permutation container: $C_6$ Parity: even Dirichlet character: $$\chi_{72}(61,\cdot)$$ Projective image: $C_1$ Projective field: Galois closure of $$\Q$$

Defining polynomial

 $f(x)$ $=$ $$x^{6} - 12x^{4} + 36x^{2} - 8$$ x^6 - 12*x^4 + 36*x^2 - 8 .

The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: $$x^{2} + 18x + 2$$

Roots:
 $r_{ 1 }$ $=$ $$14 a + 12 + \left(11 a + 10\right)\cdot 19 + \left(11 a + 9\right)\cdot 19^{2} + \left(12 a + 18\right)\cdot 19^{3} + \left(14 a + 17\right)\cdot 19^{4} +O(19^{5})$$ 14*a + 12 + (11*a + 10)*19 + (11*a + 9)*19^2 + (12*a + 18)*19^3 + (14*a + 17)*19^4+O(19^5) $r_{ 2 }$ $=$ $$18 a + 10 + 8\cdot 19 + \left(5 a + 7\right)\cdot 19^{2} + \left(a + 11\right)\cdot 19^{3} + \left(16 a + 11\right)\cdot 19^{4} +O(19^{5})$$ 18*a + 10 + 8*19 + (5*a + 7)*19^2 + (a + 11)*19^3 + (16*a + 11)*19^4+O(19^5) $r_{ 3 }$ $=$ $$15 a + 2 + \left(10 a + 2\right)\cdot 19 + \left(6 a + 2\right)\cdot 19^{2} + \left(11 a + 7\right)\cdot 19^{3} + \left(17 a + 6\right)\cdot 19^{4} +O(19^{5})$$ 15*a + 2 + (10*a + 2)*19 + (6*a + 2)*19^2 + (11*a + 7)*19^3 + (17*a + 6)*19^4+O(19^5) $r_{ 4 }$ $=$ $$5 a + 7 + \left(7 a + 8\right)\cdot 19 + \left(7 a + 9\right)\cdot 19^{2} + 6 a\cdot 19^{3} + \left(4 a + 1\right)\cdot 19^{4} +O(19^{5})$$ 5*a + 7 + (7*a + 8)*19 + (7*a + 9)*19^2 + 6*a*19^3 + (4*a + 1)*19^4+O(19^5) $r_{ 5 }$ $=$ $$a + 9 + \left(18 a + 10\right)\cdot 19 + \left(13 a + 11\right)\cdot 19^{2} + \left(17 a + 7\right)\cdot 19^{3} + \left(2 a + 7\right)\cdot 19^{4} +O(19^{5})$$ a + 9 + (18*a + 10)*19 + (13*a + 11)*19^2 + (17*a + 7)*19^3 + (2*a + 7)*19^4+O(19^5) $r_{ 6 }$ $=$ $$4 a + 17 + \left(8 a + 16\right)\cdot 19 + \left(12 a + 16\right)\cdot 19^{2} + \left(7 a + 11\right)\cdot 19^{3} + \left(a + 12\right)\cdot 19^{4} +O(19^{5})$$ 4*a + 17 + (8*a + 16)*19 + (12*a + 16)*19^2 + (7*a + 11)*19^3 + (a + 12)*19^4+O(19^5)

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.