Properties

 Label 2.243.6t5.b.b Dimension $2$ Group $S_3\times C_3$ Conductor $243$ Root number not computed Indicator $0$

Related objects

Basic invariants

 Dimension: $2$ Group: $S_3\times C_3$ Conductor: $$243$$$$\medspace = 3^{5}$$ Artin stem field: Galois closure of 6.0.177147.1 Galois orbit size: $2$ Smallest permutation container: $S_3\times C_3$ Parity: odd Determinant: 1.9.6t1.a.b Projective image: $S_3$ Projective stem field: Galois closure of 3.1.243.1

Defining polynomial

 $f(x)$ $=$ $$x^{6} - 3x^{3} + 3$$ x^6 - 3*x^3 + 3 .

The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 7.

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

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.