# Properties

 Label 2.136.8t17.a.b Dimension $2$ Group $C_4\wr C_2$ Conductor $136$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $C_4\wr C_2$ Conductor: $$136$$$$\medspace = 2^{3} \cdot 17$$ Artin stem field: 8.0.20123648.1 Galois orbit size: $2$ Smallest permutation container: $C_4\wr C_2$ Parity: odd Determinant: 1.136.4t1.b.a Projective image: $D_4$ Projective stem field: 4.2.39304.1

## Defining polynomial

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

The roots of $f$ are computed in $\Q_{ 307 }$ to precision 5.

Roots:
 $r_{ 1 }$ $=$ $$38 + 166\cdot 307 + 69\cdot 307^{2} + 158\cdot 307^{3} + 143\cdot 307^{4} +O(307^{5})$$ $r_{ 2 }$ $=$ $$50 + 75\cdot 307 + 241\cdot 307^{2} + 13\cdot 307^{3} + 63\cdot 307^{4} +O(307^{5})$$ $r_{ 3 }$ $=$ $$75 + 64\cdot 307 + 64\cdot 307^{2} + 257\cdot 307^{3} + 69\cdot 307^{4} +O(307^{5})$$ $r_{ 4 }$ $=$ $$84 + 92\cdot 307 + 186\cdot 307^{2} + 31\cdot 307^{3} + 223\cdot 307^{4} +O(307^{5})$$ $r_{ 5 }$ $=$ $$95 + 238\cdot 307 + 137\cdot 307^{2} + 64\cdot 307^{3} + 92\cdot 307^{4} +O(307^{5})$$ $r_{ 6 }$ $=$ $$148 + 267\cdot 307 + 168\cdot 307^{2} + 146\cdot 307^{3} + 65\cdot 307^{4} +O(307^{5})$$ $r_{ 7 }$ $=$ $$196 + 34\cdot 307 + 212\cdot 307^{2} + 262\cdot 307^{3} + 29\cdot 307^{4} +O(307^{5})$$ $r_{ 8 }$ $=$ $$237 + 289\cdot 307 + 147\cdot 307^{2} + 293\cdot 307^{3} + 233\cdot 307^{4} +O(307^{5})$$

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.