# Properties

 Label 2.3200.8t17.b.a Dimension $2$ Group $C_4\wr C_2$ Conductor $3200$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $C_4\wr C_2$ Conductor: $$3200$$$$\medspace = 2^{7} \cdot 5^{2}$$ Artin stem field: Galois closure of 8.0.131072000.2 Galois orbit size: $2$ Smallest permutation container: $C_4\wr C_2$ Parity: odd Determinant: 1.40.4t1.b.a Projective image: $D_4$ Projective stem field: Galois closure of 4.2.8000.1

## Defining polynomial

 $f(x)$ $=$ $$x^{8} - 6x^{6} + 12x^{4} - 10x^{2} + 5$$ x^8 - 6*x^6 + 12*x^4 - 10*x^2 + 5 .

The roots of $f$ are computed in $\Q_{ 29 }$ to precision 10.

Roots:
 $r_{ 1 }$ $=$ $$2 + 20\cdot 29 + 25\cdot 29^{2} + 17\cdot 29^{3} + 9\cdot 29^{4} + 9\cdot 29^{5} + 11\cdot 29^{6} + 12\cdot 29^{7} + 5\cdot 29^{8} + 9\cdot 29^{9} +O(29^{10})$$ 2 + 20*29 + 25*29^2 + 17*29^3 + 9*29^4 + 9*29^5 + 11*29^6 + 12*29^7 + 5*29^8 + 9*29^9+O(29^10) $r_{ 2 }$ $=$ $$3 + 23\cdot 29 + 27\cdot 29^{2} + 20\cdot 29^{3} + 28\cdot 29^{4} + 5\cdot 29^{5} + 18\cdot 29^{6} + 18\cdot 29^{7} + 11\cdot 29^{8} + 2\cdot 29^{9} +O(29^{10})$$ 3 + 23*29 + 27*29^2 + 20*29^3 + 28*29^4 + 5*29^5 + 18*29^6 + 18*29^7 + 11*29^8 + 2*29^9+O(29^10) $r_{ 3 }$ $=$ $$4 + 26\cdot 29 + 17\cdot 29^{2} + 22\cdot 29^{3} + 15\cdot 29^{4} + 19\cdot 29^{5} + 4\cdot 29^{6} + 25\cdot 29^{7} + 5\cdot 29^{8} +O(29^{10})$$ 4 + 26*29 + 17*29^2 + 22*29^3 + 15*29^4 + 19*29^5 + 4*29^6 + 25*29^7 + 5*29^8+O(29^10) $r_{ 4 }$ $=$ $$8 + 4\cdot 29 + 25\cdot 29^{2} + 7\cdot 29^{3} + 28\cdot 29^{4} + 10\cdot 29^{5} + 17\cdot 29^{6} + 10\cdot 29^{7} + 5\cdot 29^{8} +O(29^{10})$$ 8 + 4*29 + 25*29^2 + 7*29^3 + 28*29^4 + 10*29^5 + 17*29^6 + 10*29^7 + 5*29^8+O(29^10) $r_{ 5 }$ $=$ $$21 + 24\cdot 29 + 3\cdot 29^{2} + 21\cdot 29^{3} + 18\cdot 29^{5} + 11\cdot 29^{6} + 18\cdot 29^{7} + 23\cdot 29^{8} + 28\cdot 29^{9} +O(29^{10})$$ 21 + 24*29 + 3*29^2 + 21*29^3 + 18*29^5 + 11*29^6 + 18*29^7 + 23*29^8 + 28*29^9+O(29^10) $r_{ 6 }$ $=$ $$25 + 2\cdot 29 + 11\cdot 29^{2} + 6\cdot 29^{3} + 13\cdot 29^{4} + 9\cdot 29^{5} + 24\cdot 29^{6} + 3\cdot 29^{7} + 23\cdot 29^{8} + 28\cdot 29^{9} +O(29^{10})$$ 25 + 2*29 + 11*29^2 + 6*29^3 + 13*29^4 + 9*29^5 + 24*29^6 + 3*29^7 + 23*29^8 + 28*29^9+O(29^10) $r_{ 7 }$ $=$ $$26 + 5\cdot 29 + 29^{2} + 8\cdot 29^{3} + 23\cdot 29^{5} + 10\cdot 29^{6} + 10\cdot 29^{7} + 17\cdot 29^{8} + 26\cdot 29^{9} +O(29^{10})$$ 26 + 5*29 + 29^2 + 8*29^3 + 23*29^5 + 10*29^6 + 10*29^7 + 17*29^8 + 26*29^9+O(29^10) $r_{ 8 }$ $=$ $$27 + 8\cdot 29 + 3\cdot 29^{2} + 11\cdot 29^{3} + 19\cdot 29^{4} + 19\cdot 29^{5} + 17\cdot 29^{6} + 16\cdot 29^{7} + 23\cdot 29^{8} + 19\cdot 29^{9} +O(29^{10})$$ 27 + 8*29 + 3*29^2 + 11*29^3 + 19*29^4 + 19*29^5 + 17*29^6 + 16*29^7 + 23*29^8 + 19*29^9+O(29^10)

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.