# Properties

 Label 2.145.8t17.a.b Dimension $2$ Group $C_4\wr C_2$ Conductor $145$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $C_4\wr C_2$ Conductor: $$145$$$$\medspace = 5 \cdot 29$$ Artin stem field: 8.4.15243125.1 Galois orbit size: $2$ Smallest permutation container: $C_4\wr C_2$ Parity: odd Determinant: 1.145.4t1.a.b Projective image: $D_4$ Projective stem field: 4.0.121945.1

## Defining polynomial

 $f(x)$ $=$ $$x^{8} - x^{7} - 3 x^{6} + 3 x^{5} + 3 x^{4} - 6 x^{3} - 2 x^{2} + 3 x + 1$$  .

The roots of $f$ are computed in $\Q_{ 139 }$ to precision 5.

Roots:
 $r_{ 1 }$ $=$ $$10 + 35\cdot 139 + 67\cdot 139^{2} + 29\cdot 139^{3} + 61\cdot 139^{4} +O(139^{5})$$ $r_{ 2 }$ $=$ $$46 + 33\cdot 139 + 70\cdot 139^{2} + 108\cdot 139^{3} + 130\cdot 139^{4} +O(139^{5})$$ $r_{ 3 }$ $=$ $$47 + 91\cdot 139 + 134\cdot 139^{2} + 48\cdot 139^{3} + 84\cdot 139^{4} +O(139^{5})$$ $r_{ 4 }$ $=$ $$50 + 39\cdot 139 + 67\cdot 139^{2} + 129\cdot 139^{3} + 105\cdot 139^{4} +O(139^{5})$$ $r_{ 5 }$ $=$ $$51 + 11\cdot 139 + 3\cdot 139^{2} + 122\cdot 139^{3} + 53\cdot 139^{4} +O(139^{5})$$ $r_{ 6 }$ $=$ $$56 + 69\cdot 139 + 27\cdot 139^{2} + 127\cdot 139^{3} + 122\cdot 139^{4} +O(139^{5})$$ $r_{ 7 }$ $=$ $$62 + 62\cdot 139 + 133\cdot 139^{2} + 25\cdot 139^{3} + 12\cdot 139^{4} +O(139^{5})$$ $r_{ 8 }$ $=$ $$96 + 74\cdot 139 + 52\cdot 139^{2} + 103\cdot 139^{3} + 123\cdot 139^{4} +O(139^{5})$$

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.