Properties

 Label 2.278784.8t5.e.a Dimension $2$ Group $Q_8$ Conductor $278784$ Root number $1$ Indicator $-1$

Related objects

Basic invariants

 Dimension: $2$ Group: $Q_8$ Conductor: $$278784$$$$\medspace = 2^{8} \cdot 3^{2} \cdot 11^{2}$$ Frobenius-Schur indicator: $-1$ Root number: $1$ Artin field: Galois closure of 8.8.21667237072994304.1 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 $$\Q(\sqrt{6}, \sqrt{11})$$

Defining polynomial

 $f(x)$ $=$ $$x^{8} - 132x^{6} + 4356x^{4} - 30492x^{2} + 27225$$ x^8 - 132*x^6 + 4356*x^4 - 30492*x^2 + 27225 .

The roots of $f$ are computed in $\Q_{ 53 }$ to precision 10.

Roots:
 $r_{ 1 }$ $=$ $$9 + 43\cdot 53 + 21\cdot 53^{2} + 6\cdot 53^{3} + 21\cdot 53^{4} + 46\cdot 53^{5} + 40\cdot 53^{6} + 4\cdot 53^{7} + 52\cdot 53^{8} + 50\cdot 53^{9} +O(53^{10})$$ 9 + 43*53 + 21*53^2 + 6*53^3 + 21*53^4 + 46*53^5 + 40*53^6 + 4*53^7 + 52*53^8 + 50*53^9+O(53^10) $r_{ 2 }$ $=$ $$14 + 44\cdot 53 + 31\cdot 53^{2} + 37\cdot 53^{3} + 30\cdot 53^{4} + 5\cdot 53^{5} + 14\cdot 53^{6} + 47\cdot 53^{7} + 42\cdot 53^{8} + 8\cdot 53^{9} +O(53^{10})$$ 14 + 44*53 + 31*53^2 + 37*53^3 + 30*53^4 + 5*53^5 + 14*53^6 + 47*53^7 + 42*53^8 + 8*53^9+O(53^10) $r_{ 3 }$ $=$ $$17 + 7\cdot 53 + 34\cdot 53^{2} + 40\cdot 53^{3} + 15\cdot 53^{4} + 24\cdot 53^{6} + 4\cdot 53^{7} + 45\cdot 53^{8} + 33\cdot 53^{9} +O(53^{10})$$ 17 + 7*53 + 34*53^2 + 40*53^3 + 15*53^4 + 24*53^6 + 4*53^7 + 45*53^8 + 33*53^9+O(53^10) $r_{ 4 }$ $=$ $$19 + 34\cdot 53 + 45\cdot 53^{2} + 31\cdot 53^{3} + 10\cdot 53^{4} + 6\cdot 53^{5} + 18\cdot 53^{6} + 34\cdot 53^{7} + 32\cdot 53^{8} + 11\cdot 53^{9} +O(53^{10})$$ 19 + 34*53 + 45*53^2 + 31*53^3 + 10*53^4 + 6*53^5 + 18*53^6 + 34*53^7 + 32*53^8 + 11*53^9+O(53^10) $r_{ 5 }$ $=$ $$34 + 18\cdot 53 + 7\cdot 53^{2} + 21\cdot 53^{3} + 42\cdot 53^{4} + 46\cdot 53^{5} + 34\cdot 53^{6} + 18\cdot 53^{7} + 20\cdot 53^{8} + 41\cdot 53^{9} +O(53^{10})$$ 34 + 18*53 + 7*53^2 + 21*53^3 + 42*53^4 + 46*53^5 + 34*53^6 + 18*53^7 + 20*53^8 + 41*53^9+O(53^10) $r_{ 6 }$ $=$ $$36 + 45\cdot 53 + 18\cdot 53^{2} + 12\cdot 53^{3} + 37\cdot 53^{4} + 52\cdot 53^{5} + 28\cdot 53^{6} + 48\cdot 53^{7} + 7\cdot 53^{8} + 19\cdot 53^{9} +O(53^{10})$$ 36 + 45*53 + 18*53^2 + 12*53^3 + 37*53^4 + 52*53^5 + 28*53^6 + 48*53^7 + 7*53^8 + 19*53^9+O(53^10) $r_{ 7 }$ $=$ $$39 + 8\cdot 53 + 21\cdot 53^{2} + 15\cdot 53^{3} + 22\cdot 53^{4} + 47\cdot 53^{5} + 38\cdot 53^{6} + 5\cdot 53^{7} + 10\cdot 53^{8} + 44\cdot 53^{9} +O(53^{10})$$ 39 + 8*53 + 21*53^2 + 15*53^3 + 22*53^4 + 47*53^5 + 38*53^6 + 5*53^7 + 10*53^8 + 44*53^9+O(53^10) $r_{ 8 }$ $=$ $$44 + 9\cdot 53 + 31\cdot 53^{2} + 46\cdot 53^{3} + 31\cdot 53^{4} + 6\cdot 53^{5} + 12\cdot 53^{6} + 48\cdot 53^{7} + 2\cdot 53^{9} +O(53^{10})$$ 44 + 9*53 + 31*53^2 + 46*53^3 + 31*53^4 + 6*53^5 + 12*53^6 + 48*53^7 + 2*53^9+O(53^10)

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.