# Properties

 Label 2.112896.8t5.g.a Dimension $2$ Group $Q_8$ Conductor $112896$ Root number $-1$ Indicator $-1$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $Q_8$ Conductor: $$112896$$$$\medspace = 2^{8} \cdot 3^{2} \cdot 7^{2}$$ Frobenius-Schur indicator: $-1$ Root number: $-1$ Artin field: Galois closure of 8.8.359729184374784.2 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 $$\Q(\sqrt{6}, \sqrt{14})$$

## Defining polynomial

 $f(x)$ $=$ $$x^{8} - 84x^{6} + 1890x^{4} - 10584x^{2} + 1764$$ x^8 - 84*x^6 + 1890*x^4 - 10584*x^2 + 1764 .

The roots of $f$ are computed in $\Q_{ 67 }$ to precision 10.

Roots:
 $r_{ 1 }$ $=$ $$13 + 62\cdot 67 + 27\cdot 67^{2} + 51\cdot 67^{3} + 5\cdot 67^{4} + 66\cdot 67^{5} + 36\cdot 67^{6} + 40\cdot 67^{7} + 3\cdot 67^{8} + 45\cdot 67^{9} +O(67^{10})$$ 13 + 62*67 + 27*67^2 + 51*67^3 + 5*67^4 + 66*67^5 + 36*67^6 + 40*67^7 + 3*67^8 + 45*67^9+O(67^10) $r_{ 2 }$ $=$ $$16 + 9\cdot 67 + 27\cdot 67^{2} + 2\cdot 67^{3} + 44\cdot 67^{4} + 14\cdot 67^{5} + 46\cdot 67^{6} + 25\cdot 67^{7} + 40\cdot 67^{8} + 63\cdot 67^{9} +O(67^{10})$$ 16 + 9*67 + 27*67^2 + 2*67^3 + 44*67^4 + 14*67^5 + 46*67^6 + 25*67^7 + 40*67^8 + 63*67^9+O(67^10) $r_{ 3 }$ $=$ $$17 + 48\cdot 67 + 66\cdot 67^{2} + 53\cdot 67^{3} + 55\cdot 67^{4} + 16\cdot 67^{5} + 49\cdot 67^{6} + 40\cdot 67^{7} + 15\cdot 67^{8} + 26\cdot 67^{9} +O(67^{10})$$ 17 + 48*67 + 66*67^2 + 53*67^3 + 55*67^4 + 16*67^5 + 49*67^6 + 40*67^7 + 15*67^8 + 26*67^9+O(67^10) $r_{ 4 }$ $=$ $$24 + 7\cdot 67 + 65\cdot 67^{2} + 14\cdot 67^{3} + 63\cdot 67^{4} + 32\cdot 67^{5} + 44\cdot 67^{6} + 56\cdot 67^{7} + 54\cdot 67^{8} + 21\cdot 67^{9} +O(67^{10})$$ 24 + 7*67 + 65*67^2 + 14*67^3 + 63*67^4 + 32*67^5 + 44*67^6 + 56*67^7 + 54*67^8 + 21*67^9+O(67^10) $r_{ 5 }$ $=$ $$43 + 59\cdot 67 + 67^{2} + 52\cdot 67^{3} + 3\cdot 67^{4} + 34\cdot 67^{5} + 22\cdot 67^{6} + 10\cdot 67^{7} + 12\cdot 67^{8} + 45\cdot 67^{9} +O(67^{10})$$ 43 + 59*67 + 67^2 + 52*67^3 + 3*67^4 + 34*67^5 + 22*67^6 + 10*67^7 + 12*67^8 + 45*67^9+O(67^10) $r_{ 6 }$ $=$ $$50 + 18\cdot 67 + 13\cdot 67^{3} + 11\cdot 67^{4} + 50\cdot 67^{5} + 17\cdot 67^{6} + 26\cdot 67^{7} + 51\cdot 67^{8} + 40\cdot 67^{9} +O(67^{10})$$ 50 + 18*67 + 13*67^3 + 11*67^4 + 50*67^5 + 17*67^6 + 26*67^7 + 51*67^8 + 40*67^9+O(67^10) $r_{ 7 }$ $=$ $$51 + 57\cdot 67 + 39\cdot 67^{2} + 64\cdot 67^{3} + 22\cdot 67^{4} + 52\cdot 67^{5} + 20\cdot 67^{6} + 41\cdot 67^{7} + 26\cdot 67^{8} + 3\cdot 67^{9} +O(67^{10})$$ 51 + 57*67 + 39*67^2 + 64*67^3 + 22*67^4 + 52*67^5 + 20*67^6 + 41*67^7 + 26*67^8 + 3*67^9+O(67^10) $r_{ 8 }$ $=$ $$54 + 4\cdot 67 + 39\cdot 67^{2} + 15\cdot 67^{3} + 61\cdot 67^{4} + 30\cdot 67^{6} + 26\cdot 67^{7} + 63\cdot 67^{8} + 21\cdot 67^{9} +O(67^{10})$$ 54 + 4*67 + 39*67^2 + 15*67^3 + 61*67^4 + 30*67^6 + 26*67^7 + 63*67^8 + 21*67^9+O(67^10)

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.