# Properties

 Label 2.120.8t11.c.b Dimension 2 Group $Q_8:C_2$ Conductor $2^{3} \cdot 3 \cdot 5$ Root number not computed Frobenius-Schur indicator 0

# Related objects

## Basic invariants

 Dimension: $2$ Group: $Q_8:C_2$ Conductor: $120= 2^{3} \cdot 3 \cdot 5$ Artin number field: Splitting field of $f= x^{8} - 3 x^{7} + 3 x^{6} + x^{5} - 2 x^{4} - 3 x^{3} + 7 x^{2} - 4 x + 1$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $Q_8:C_2$ Parity: Odd Determinant: 1.120.2t1.b.a

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 79 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $7 + 19\cdot 79 + 44\cdot 79^{2} + 40\cdot 79^{3} + 47\cdot 79^{4} +O\left(79^{ 5 }\right)$ $r_{ 2 }$ $=$ $17 + 74\cdot 79 + 66\cdot 79^{2} + 61\cdot 79^{3} + 63\cdot 79^{4} +O\left(79^{ 5 }\right)$ $r_{ 3 }$ $=$ $19 + 78\cdot 79 + 12\cdot 79^{2} + 15\cdot 79^{3} + 56\cdot 79^{4} +O\left(79^{ 5 }\right)$ $r_{ 4 }$ $=$ $32 + 66\cdot 79 + 21\cdot 79^{2} + 70\cdot 79^{3} + 68\cdot 79^{4} +O\left(79^{ 5 }\right)$ $r_{ 5 }$ $=$ $33 + 44\cdot 79 + 78\cdot 79^{2} + 65\cdot 79^{3} + 37\cdot 79^{4} +O\left(79^{ 5 }\right)$ $r_{ 6 }$ $=$ $38 + 72\cdot 79 + 68\cdot 79^{2} + 13\cdot 79^{3} + 12\cdot 79^{4} +O\left(79^{ 5 }\right)$ $r_{ 7 }$ $=$ $40 + 45\cdot 79 + 63\cdot 79^{2} + 67\cdot 79^{3} + 17\cdot 79^{4} +O\left(79^{ 5 }\right)$ $r_{ 8 }$ $=$ $54 + 73\cdot 79 + 37\cdot 79^{2} + 59\cdot 79^{3} + 11\cdot 79^{4} +O\left(79^{ 5 }\right)$

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

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

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 8 }$ Character value $1$ $1$ $()$ $2$ $1$ $2$ $(1,6)(2,8)(3,7)(4,5)$ $-2$ $2$ $2$ $(3,7)(4,5)$ $0$ $2$ $2$ $(1,4)(2,7)(3,8)(5,6)$ $0$ $2$ $2$ $(1,3)(2,4)(5,8)(6,7)$ $0$ $1$ $4$ $(1,8,6,2)(3,5,7,4)$ $2 \zeta_{4}$ $1$ $4$ $(1,2,6,8)(3,4,7,5)$ $-2 \zeta_{4}$ $2$ $4$ $(1,4,6,5)(2,7,8,3)$ $0$ $2$ $4$ $(1,3,6,7)(2,4,8,5)$ $0$ $2$ $4$ $(1,8,6,2)(3,4,7,5)$ $0$
The blue line marks the conjugacy class containing complex conjugation.