# Properties

 Label 2.93025.8t5.a.a Dimension $2$ Group $Q_8$ Conductor $93025$ Root number $-1$ Indicator $-1$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $Q_8$ Conductor: $$93025$$$$\medspace = 5^{2} \cdot 61^{2}$$ Frobenius-Schur indicator: $-1$ Root number: $-1$ Artin field: Galois closure of 8.0.805005849390625.1 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 $$\Q(\sqrt{5}, \sqrt{61})$$

## Defining polynomial

 $f(x)$ $=$ $$x^{8} - 3x^{7} + 57x^{6} + 135x^{5} + 306x^{4} + 5365x^{3} + 23383x^{2} + 40951x + 75421$$ x^8 - 3*x^7 + 57*x^6 + 135*x^5 + 306*x^4 + 5365*x^3 + 23383*x^2 + 40951*x + 75421 .

The roots of $f$ are computed in $\Q_{ 199 }$ to precision 5.

Roots:
 $r_{ 1 }$ $=$ $$106\cdot 199 + 78\cdot 199^{2} + 61\cdot 199^{3} + 63\cdot 199^{4} +O(199^{5})$$ 106*199 + 78*199^2 + 61*199^3 + 63*199^4+O(199^5) $r_{ 2 }$ $=$ $$3 + 25\cdot 199 + 148\cdot 199^{2} + 111\cdot 199^{3} + 58\cdot 199^{4} +O(199^{5})$$ 3 + 25*199 + 148*199^2 + 111*199^3 + 58*199^4+O(199^5) $r_{ 3 }$ $=$ $$18 + 179\cdot 199 + 8\cdot 199^{2} + 14\cdot 199^{3} + 187\cdot 199^{4} +O(199^{5})$$ 18 + 179*199 + 8*199^2 + 14*199^3 + 187*199^4+O(199^5) $r_{ 4 }$ $=$ $$30 + 41\cdot 199 + 63\cdot 199^{2} + 50\cdot 199^{3} + 23\cdot 199^{4} +O(199^{5})$$ 30 + 41*199 + 63*199^2 + 50*199^3 + 23*199^4+O(199^5) $r_{ 5 }$ $=$ $$79 + 131\cdot 199 + 171\cdot 199^{2} + 154\cdot 199^{3} + 177\cdot 199^{4} +O(199^{5})$$ 79 + 131*199 + 171*199^2 + 154*199^3 + 177*199^4+O(199^5) $r_{ 6 }$ $=$ $$118 + 37\cdot 199 + 58\cdot 199^{2} + 80\cdot 199^{3} + 74\cdot 199^{4} +O(199^{5})$$ 118 + 37*199 + 58*199^2 + 80*199^3 + 74*199^4+O(199^5) $r_{ 7 }$ $=$ $$157 + 72\cdot 199 + 156\cdot 199^{2} + 13\cdot 199^{3} + 121\cdot 199^{4} +O(199^{5})$$ 157 + 72*199 + 156*199^2 + 13*199^3 + 121*199^4+O(199^5) $r_{ 8 }$ $=$ $$195 + 3\cdot 199 + 111\cdot 199^{2} + 110\cdot 199^{3} + 90\cdot 199^{4} +O(199^{5})$$ 195 + 3*199 + 111*199^2 + 110*199^3 + 90*199^4+O(199^5)

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.