# Properties

 Label 2.73984.8t5.a.a Dimension $2$ Group $Q_8$ Conductor $73984$ Root number $-1$ Indicator $-1$

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## Basic invariants

 Dimension: $2$ Group: $Q_8$ Conductor: $$73984$$$$\medspace = 2^{8} \cdot 17^{2}$$ Frobenius-Schur indicator: $-1$ Root number: $-1$ Artin field: Galois closure of 8.8.101240302206976.2 Galois orbit size: $1$ Smallest permutation container: $Q_8$ Parity: even Determinant: 1.1.1t1.a.a Projective image: $C_2^2$ Projective field: Galois closure of $$\Q(\sqrt{2}, \sqrt{17})$$

## Defining polynomial

 $f(x)$ $=$ $$x^{8} - 68x^{6} + 986x^{4} - 4624x^{2} + 4624$$ x^8 - 68*x^6 + 986*x^4 - 4624*x^2 + 4624 .

The roots of $f$ are computed in $\Q_{ 137 }$ to precision 10.

Roots:
 $r_{ 1 }$ $=$ $$2 + 15\cdot 137 + 21\cdot 137^{2} + 110\cdot 137^{3} + 90\cdot 137^{4} + 107\cdot 137^{5} + 129\cdot 137^{6} + 25\cdot 137^{7} + 64\cdot 137^{8} + 103\cdot 137^{9} +O(137^{10})$$ 2 + 15*137 + 21*137^2 + 110*137^3 + 90*137^4 + 107*137^5 + 129*137^6 + 25*137^7 + 64*137^8 + 103*137^9+O(137^10) $r_{ 2 }$ $=$ $$3 + 71\cdot 137 + 26\cdot 137^{2} + 137^{3} + 82\cdot 137^{4} + 23\cdot 137^{5} + 17\cdot 137^{6} + 50\cdot 137^{7} + 131\cdot 137^{8} + 37\cdot 137^{9} +O(137^{10})$$ 3 + 71*137 + 26*137^2 + 137^3 + 82*137^4 + 23*137^5 + 17*137^6 + 50*137^7 + 131*137^8 + 37*137^9+O(137^10) $r_{ 3 }$ $=$ $$47 + 121\cdot 137 + 22\cdot 137^{2} + 28\cdot 137^{3} + 44\cdot 137^{4} + 95\cdot 137^{5} + 66\cdot 137^{6} + 103\cdot 137^{7} + 55\cdot 137^{8} + 65\cdot 137^{9} +O(137^{10})$$ 47 + 121*137 + 22*137^2 + 28*137^3 + 44*137^4 + 95*137^5 + 66*137^6 + 103*137^7 + 55*137^8 + 65*137^9+O(137^10) $r_{ 4 }$ $=$ $$60 + 136\cdot 137 + 94\cdot 137^{2} + 99\cdot 137^{3} + 84\cdot 137^{4} + 63\cdot 137^{5} + 52\cdot 137^{6} + 70\cdot 137^{7} + 50\cdot 137^{8} + 33\cdot 137^{9} +O(137^{10})$$ 60 + 136*137 + 94*137^2 + 99*137^3 + 84*137^4 + 63*137^5 + 52*137^6 + 70*137^7 + 50*137^8 + 33*137^9+O(137^10) $r_{ 5 }$ $=$ $$77 + 42\cdot 137^{2} + 37\cdot 137^{3} + 52\cdot 137^{4} + 73\cdot 137^{5} + 84\cdot 137^{6} + 66\cdot 137^{7} + 86\cdot 137^{8} + 103\cdot 137^{9} +O(137^{10})$$ 77 + 42*137^2 + 37*137^3 + 52*137^4 + 73*137^5 + 84*137^6 + 66*137^7 + 86*137^8 + 103*137^9+O(137^10) $r_{ 6 }$ $=$ $$90 + 15\cdot 137 + 114\cdot 137^{2} + 108\cdot 137^{3} + 92\cdot 137^{4} + 41\cdot 137^{5} + 70\cdot 137^{6} + 33\cdot 137^{7} + 81\cdot 137^{8} + 71\cdot 137^{9} +O(137^{10})$$ 90 + 15*137 + 114*137^2 + 108*137^3 + 92*137^4 + 41*137^5 + 70*137^6 + 33*137^7 + 81*137^8 + 71*137^9+O(137^10) $r_{ 7 }$ $=$ $$134 + 65\cdot 137 + 110\cdot 137^{2} + 135\cdot 137^{3} + 54\cdot 137^{4} + 113\cdot 137^{5} + 119\cdot 137^{6} + 86\cdot 137^{7} + 5\cdot 137^{8} + 99\cdot 137^{9} +O(137^{10})$$ 134 + 65*137 + 110*137^2 + 135*137^3 + 54*137^4 + 113*137^5 + 119*137^6 + 86*137^7 + 5*137^8 + 99*137^9+O(137^10) $r_{ 8 }$ $=$ $$135 + 121\cdot 137 + 115\cdot 137^{2} + 26\cdot 137^{3} + 46\cdot 137^{4} + 29\cdot 137^{5} + 7\cdot 137^{6} + 111\cdot 137^{7} + 72\cdot 137^{8} + 33\cdot 137^{9} +O(137^{10})$$ 135 + 121*137 + 115*137^2 + 26*137^3 + 46*137^4 + 29*137^5 + 7*137^6 + 111*137^7 + 72*137^8 + 33*137^9+O(137^10)

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.