# Properties

 Label 2.127449.8t5.b.a Dimension $2$ Group $Q_8$ Conductor $127449$ Root number $1$ Indicator $-1$

# Learn more about

## Basic invariants

 Dimension: $2$ Group: $Q_8$ Conductor: $$127449$$$$\medspace = 3^{2} \cdot 7^{2} \cdot 17^{2}$$ Frobenius-Schur indicator: $-1$ Root number: $1$ Artin number field: Galois closure of 8.0.2070185663499849.1 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 $$\Q(\sqrt{17}, \sqrt{21})$$

## Defining polynomial

 $f(x)$ $=$ $x^{8} - x^{7} + 65 x^{6} - 439 x^{5} + 1876 x^{4} - 12191 x^{3} + 60887 x^{2} - 124718 x + 121291$.

The roots of $f$ are computed in $\Q_{ 89 }$ to precision 5.

Roots:
 $r_{ 1 }$ $=$ $17 + 59\cdot 89 + 69\cdot 89^{2} + 71\cdot 89^{3} + 27\cdot 89^{4} +O\left(89^{ 5 }\right)$ $r_{ 2 }$ $=$ $19 + 75\cdot 89 + 57\cdot 89^{2} + 78\cdot 89^{3} + 11\cdot 89^{4} +O\left(89^{ 5 }\right)$ $r_{ 3 }$ $=$ $24 + 65\cdot 89 + 57\cdot 89^{2} + 13\cdot 89^{3} + 44\cdot 89^{4} +O\left(89^{ 5 }\right)$ $r_{ 4 }$ $=$ $29 + 66\cdot 89 + 82\cdot 89^{2} + 11\cdot 89^{3} + 10\cdot 89^{4} +O\left(89^{ 5 }\right)$ $r_{ 5 }$ $=$ $33 + 24\cdot 89 + 11\cdot 89^{2} + 31\cdot 89^{3} + 12\cdot 89^{4} +O\left(89^{ 5 }\right)$ $r_{ 6 }$ $=$ $67 + 82\cdot 89 + 53\cdot 89^{2} + 29\cdot 89^{3} + 20\cdot 89^{4} +O\left(89^{ 5 }\right)$ $r_{ 7 }$ $=$ $82 + 66\cdot 89 + 31\cdot 89^{2} + 46\cdot 89^{3} + 19\cdot 89^{4} +O\left(89^{ 5 }\right)$ $r_{ 8 }$ $=$ $86 + 4\cdot 89 + 80\cdot 89^{2} + 72\cdot 89^{3} + 31\cdot 89^{4} +O\left(89^{ 5 }\right)$

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.