# Properties

 Label 2.416.8t17.a.a Dimension $2$ Group $C_4\wr C_2$ Conductor $416$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $C_4\wr C_2$ Conductor: $$416$$$$\medspace = 2^{5} \cdot 13$$ Artin stem field: Galois closure of 8.0.143982592.1 Galois orbit size: $2$ Smallest permutation container: $C_4\wr C_2$ Parity: odd Determinant: 1.13.4t1.a.a Projective image: $D_4$ Projective stem field: Galois closure of 4.2.35152.1

## Defining polynomial

 $f(x)$ $=$ $$x^{8} + 6x^{4} + 13$$ x^8 + 6*x^4 + 13 .

The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: $$x^{2} + 21x + 5$$

Roots:
 $r_{ 1 }$ $=$ $$15 a + 5 + \left(9 a + 12\right)\cdot 23 + \left(5 a + 6\right)\cdot 23^{2} + \left(21 a + 14\right)\cdot 23^{3} + \left(9 a + 18\right)\cdot 23^{4} + \left(19 a + 17\right)\cdot 23^{5} + \left(4 a + 12\right)\cdot 23^{6} + \left(3 a + 17\right)\cdot 23^{7} + \left(8 a + 6\right)\cdot 23^{8} + 20\cdot 23^{9} +O(23^{10})$$ 15*a + 5 + (9*a + 12)*23 + (5*a + 6)*23^2 + (21*a + 14)*23^3 + (9*a + 18)*23^4 + (19*a + 17)*23^5 + (4*a + 12)*23^6 + (3*a + 17)*23^7 + (8*a + 6)*23^8 + 20*23^9+O(23^10) $r_{ 2 }$ $=$ $$10 a + 20 + \left(17 a + 4\right)\cdot 23 + \left(10 a + 3\right)\cdot 23^{2} + \left(15 a + 13\right)\cdot 23^{3} + \left(6 a + 5\right)\cdot 23^{4} + \left(14 a + 10\right)\cdot 23^{5} + \left(2 a + 19\right)\cdot 23^{6} + \left(7 a + 3\right)\cdot 23^{7} + \left(18 a + 11\right)\cdot 23^{8} + \left(14 a + 10\right)\cdot 23^{9} +O(23^{10})$$ 10*a + 20 + (17*a + 4)*23 + (10*a + 3)*23^2 + (15*a + 13)*23^3 + (6*a + 5)*23^4 + (14*a + 10)*23^5 + (2*a + 19)*23^6 + (7*a + 3)*23^7 + (18*a + 11)*23^8 + (14*a + 10)*23^9+O(23^10) $r_{ 3 }$ $=$ $$10 a + 6 + \left(17 a + 16\right)\cdot 23 + \left(10 a + 15\right)\cdot 23^{2} + \left(15 a + 12\right)\cdot 23^{3} + \left(6 a + 19\right)\cdot 23^{4} + \left(14 a + 13\right)\cdot 23^{5} + \left(2 a + 12\right)\cdot 23^{6} + \left(7 a + 7\right)\cdot 23^{7} + \left(18 a + 5\right)\cdot 23^{8} + \left(14 a + 1\right)\cdot 23^{9} +O(23^{10})$$ 10*a + 6 + (17*a + 16)*23 + (10*a + 15)*23^2 + (15*a + 12)*23^3 + (6*a + 19)*23^4 + (14*a + 13)*23^5 + (2*a + 12)*23^6 + (7*a + 7)*23^7 + (18*a + 5)*23^8 + (14*a + 1)*23^9+O(23^10) $r_{ 4 }$ $=$ $$15 a + 11 + \left(9 a + 6\right)\cdot 23 + \left(5 a + 15\right)\cdot 23^{2} + \left(21 a + 17\right)\cdot 23^{3} + \left(9 a + 5\right)\cdot 23^{4} + \left(19 a + 22\right)\cdot 23^{5} + \left(4 a + 19\right)\cdot 23^{6} + \left(3 a + 3\right)\cdot 23^{7} + \left(8 a + 3\right)\cdot 23^{8} + 10\cdot 23^{9} +O(23^{10})$$ 15*a + 11 + (9*a + 6)*23 + (5*a + 15)*23^2 + (21*a + 17)*23^3 + (9*a + 5)*23^4 + (19*a + 22)*23^5 + (4*a + 19)*23^6 + (3*a + 3)*23^7 + (8*a + 3)*23^8 + 10*23^9+O(23^10) $r_{ 5 }$ $=$ $$8 a + 18 + \left(13 a + 10\right)\cdot 23 + \left(17 a + 16\right)\cdot 23^{2} + \left(a + 8\right)\cdot 23^{3} + \left(13 a + 4\right)\cdot 23^{4} + \left(3 a + 5\right)\cdot 23^{5} + \left(18 a + 10\right)\cdot 23^{6} + \left(19 a + 5\right)\cdot 23^{7} + \left(14 a + 16\right)\cdot 23^{8} + \left(22 a + 2\right)\cdot 23^{9} +O(23^{10})$$ 8*a + 18 + (13*a + 10)*23 + (17*a + 16)*23^2 + (a + 8)*23^3 + (13*a + 4)*23^4 + (3*a + 5)*23^5 + (18*a + 10)*23^6 + (19*a + 5)*23^7 + (14*a + 16)*23^8 + (22*a + 2)*23^9+O(23^10) $r_{ 6 }$ $=$ $$13 a + 3 + \left(5 a + 18\right)\cdot 23 + \left(12 a + 19\right)\cdot 23^{2} + \left(7 a + 9\right)\cdot 23^{3} + \left(16 a + 17\right)\cdot 23^{4} + \left(8 a + 12\right)\cdot 23^{5} + \left(20 a + 3\right)\cdot 23^{6} + \left(15 a + 19\right)\cdot 23^{7} + \left(4 a + 11\right)\cdot 23^{8} + \left(8 a + 12\right)\cdot 23^{9} +O(23^{10})$$ 13*a + 3 + (5*a + 18)*23 + (12*a + 19)*23^2 + (7*a + 9)*23^3 + (16*a + 17)*23^4 + (8*a + 12)*23^5 + (20*a + 3)*23^6 + (15*a + 19)*23^7 + (4*a + 11)*23^8 + (8*a + 12)*23^9+O(23^10) $r_{ 7 }$ $=$ $$13 a + 17 + \left(5 a + 6\right)\cdot 23 + \left(12 a + 7\right)\cdot 23^{2} + \left(7 a + 10\right)\cdot 23^{3} + \left(16 a + 3\right)\cdot 23^{4} + \left(8 a + 9\right)\cdot 23^{5} + \left(20 a + 10\right)\cdot 23^{6} + \left(15 a + 15\right)\cdot 23^{7} + \left(4 a + 17\right)\cdot 23^{8} + \left(8 a + 21\right)\cdot 23^{9} +O(23^{10})$$ 13*a + 17 + (5*a + 6)*23 + (12*a + 7)*23^2 + (7*a + 10)*23^3 + (16*a + 3)*23^4 + (8*a + 9)*23^5 + (20*a + 10)*23^6 + (15*a + 15)*23^7 + (4*a + 17)*23^8 + (8*a + 21)*23^9+O(23^10) $r_{ 8 }$ $=$ $$8 a + 12 + \left(13 a + 16\right)\cdot 23 + \left(17 a + 7\right)\cdot 23^{2} + \left(a + 5\right)\cdot 23^{3} + \left(13 a + 17\right)\cdot 23^{4} + 3 a\cdot 23^{5} + \left(18 a + 3\right)\cdot 23^{6} + \left(19 a + 19\right)\cdot 23^{7} + \left(14 a + 19\right)\cdot 23^{8} + \left(22 a + 12\right)\cdot 23^{9} +O(23^{10})$$ 8*a + 12 + (13*a + 16)*23 + (17*a + 7)*23^2 + (a + 5)*23^3 + (13*a + 17)*23^4 + 3*a*23^5 + (18*a + 3)*23^6 + (19*a + 19)*23^7 + (14*a + 19)*23^8 + (22*a + 12)*23^9+O(23^10)

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.