Properties

 Label 2.68.8t17.a.b Dimension $2$ Group $C_4\wr C_2$ Conductor $68$ Root number not computed Indicator $0$

Related objects

Basic invariants

 Dimension: $2$ Group: $C_4\wr C_2$ Conductor: $$68$$$$\medspace = 2^{2} \cdot 17$$ Artin stem field: Galois closure of 8.0.1257728.1 Galois orbit size: $2$ Smallest permutation container: $C_4\wr C_2$ Parity: odd Determinant: 1.68.4t1.a.a Projective image: $D_4$ Projective stem field: Galois closure of 4.2.19652.1

Defining polynomial

 $f(x)$ $=$ $$x^{8} - 2x^{7} + 4x^{5} - 4x^{4} + 3x^{2} - 2x + 1$$ x^8 - 2*x^7 + 4*x^5 - 4*x^4 + 3*x^2 - 2*x + 1 .

The roots of $f$ are computed in $\Q_{ 149 }$ to precision 5.

Roots:
 $r_{ 1 }$ $=$ $$23 + 19\cdot 149 + 46\cdot 149^{2} + 81\cdot 149^{3} + 103\cdot 149^{4} +O(149^{5})$$ 23 + 19*149 + 46*149^2 + 81*149^3 + 103*149^4+O(149^5) $r_{ 2 }$ $=$ $$61 + 34\cdot 149 + 129\cdot 149^{2} + 79\cdot 149^{3} + 110\cdot 149^{4} +O(149^{5})$$ 61 + 34*149 + 129*149^2 + 79*149^3 + 110*149^4+O(149^5) $r_{ 3 }$ $=$ $$81 + 8\cdot 149 + 91\cdot 149^{2} + 83\cdot 149^{3} + 71\cdot 149^{4} +O(149^{5})$$ 81 + 8*149 + 91*149^2 + 83*149^3 + 71*149^4+O(149^5) $r_{ 4 }$ $=$ $$98 + 18\cdot 149 + 67\cdot 149^{3} + 149^{4} +O(149^{5})$$ 98 + 18*149 + 67*149^3 + 149^4+O(149^5) $r_{ 5 }$ $=$ $$100 + 63\cdot 149 + 100\cdot 149^{2} + 70\cdot 149^{3} + 78\cdot 149^{4} +O(149^{5})$$ 100 + 63*149 + 100*149^2 + 70*149^3 + 78*149^4+O(149^5) $r_{ 6 }$ $=$ $$122 + 35\cdot 149 + 36\cdot 149^{2} + 22\cdot 149^{3} + 101\cdot 149^{4} +O(149^{5})$$ 122 + 35*149 + 36*149^2 + 22*149^3 + 101*149^4+O(149^5) $r_{ 7 }$ $=$ $$124 + 114\cdot 149 + 49\cdot 149^{2} + 142\cdot 149^{3} + 36\cdot 149^{4} +O(149^{5})$$ 124 + 114*149 + 49*149^2 + 142*149^3 + 36*149^4+O(149^5) $r_{ 8 }$ $=$ $$138 + 2\cdot 149 + 143\cdot 149^{2} + 48\cdot 149^{3} + 92\cdot 149^{4} +O(149^{5})$$ 138 + 2*149 + 143*149^2 + 48*149^3 + 92*149^4+O(149^5)

Generators of the action on the roots $r_1, \ldots, r_{ 8 }$

 Cycle notation $(1,8,4,7)(2,6,3,5)$ $(2,3)(7,8)$ $(1,4)(2,3)(5,6)(7,8)$ $(1,5,4,6)(2,7,3,8)$ $(2,7,3,8)$

Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 8 }$ Character value $1$ $1$ $()$ $2$ $1$ $2$ $(1,4)(2,3)(5,6)(7,8)$ $-2$ $2$ $2$ $(2,3)(7,8)$ $0$ $4$ $2$ $(1,2)(3,4)(5,7)(6,8)$ $0$ $1$ $4$ $(1,5,4,6)(2,7,3,8)$ $2 \zeta_{4}$ $1$ $4$ $(1,6,4,5)(2,8,3,7)$ $-2 \zeta_{4}$ $2$ $4$ $(2,7,3,8)$ $\zeta_{4} + 1$ $2$ $4$ $(2,8,3,7)$ $-\zeta_{4} + 1$ $2$ $4$ $(1,5,4,6)(2,3)(7,8)$ $\zeta_{4} - 1$ $2$ $4$ $(1,6,4,5)(2,3)(7,8)$ $-\zeta_{4} - 1$ $2$ $4$ $(1,5,4,6)(2,8,3,7)$ $0$ $4$ $4$ $(1,8,4,7)(2,6,3,5)$ $0$ $4$ $8$ $(1,2,6,8,4,3,5,7)$ $0$ $4$ $8$ $(1,8,5,2,4,7,6,3)$ $0$

The blue line marks the conjugacy class containing complex conjugation.