Properties

 Label 2.21025.8t5.a Dimension 2 Group $Q_8$ Conductor $5^{2} \cdot 29^{2}$ Frobenius-Schur indicator -1

Related objects

Basic invariants

 Dimension: $2$ Group: $Q_8$ Conductor: $21025= 5^{2} \cdot 29^{2}$ Artin number field: Splitting field of $f= x^{8} - x^{7} - 47 x^{6} + 40 x^{5} + 581 x^{4} - 220 x^{3} - 2038 x^{2} - 932 x - 109$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $Q_8$ Parity: Even Projective image: $C_2^2$ Projective field: Galois closure of $$\Q(\sqrt{5}, \sqrt{29})$$

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 59 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $2 + 7\cdot 59 + 51\cdot 59^{2} + 13\cdot 59^{3} + 4\cdot 59^{4} +O\left(59^{ 5 }\right)$ $r_{ 2 }$ $=$ $3 + 27\cdot 59 + 36\cdot 59^{2} + 42\cdot 59^{3} + 34\cdot 59^{4} +O\left(59^{ 5 }\right)$ $r_{ 3 }$ $=$ $4 + 38\cdot 59 + 40\cdot 59^{2} + 43\cdot 59^{3} + 49\cdot 59^{4} +O\left(59^{ 5 }\right)$ $r_{ 4 }$ $=$ $9 + 55\cdot 59 + 59^{2} + 31\cdot 59^{3} + 42\cdot 59^{4} +O\left(59^{ 5 }\right)$ $r_{ 5 }$ $=$ $12 + 30\cdot 59 + 19\cdot 59^{2} + 48\cdot 59^{3} + 3\cdot 59^{4} +O\left(59^{ 5 }\right)$ $r_{ 6 }$ $=$ $19 + 23\cdot 59 + 57\cdot 59^{2} + 42\cdot 59^{3} + 43\cdot 59^{4} +O\left(59^{ 5 }\right)$ $r_{ 7 }$ $=$ $28 + 42\cdot 59 + 39\cdot 59^{2} + 35\cdot 59^{3} + 41\cdot 59^{4} +O\left(59^{ 5 }\right)$ $r_{ 8 }$ $=$ $42 + 12\cdot 59 + 48\cdot 59^{2} + 36\cdot 59^{3} + 15\cdot 59^{4} +O\left(59^{ 5 }\right)$

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

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

Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 8 }$ Character values $c1$ $1$ $1$ $()$ $2$ $1$ $2$ $(1,3)(2,5)(4,6)(7,8)$ $-2$ $2$ $4$ $(1,7,3,8)(2,4,5,6)$ $0$ $2$ $4$ $(1,4,3,6)(2,8,5,7)$ $0$ $2$ $4$ $(1,2,3,5)(4,7,6,8)$ $0$
The blue line marks the conjugacy class containing complex conjugation.