Properties

 Label 2.95.8t6.a.b Dimension 2 Group $D_{8}$ Conductor $5 \cdot 19$ Root number 1 Frobenius-Schur indicator 1

Related objects

Basic invariants

 Dimension: $2$ Group: $D_{8}$ Conductor: $95= 5 \cdot 19$ Artin number field: Splitting field of $f= x^{8} - x^{7} + x^{5} - 2 x^{4} - x^{3} + 2 x^{2} + 2 x - 1$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $D_{8}$ Parity: Odd Determinant: 1.95.2t1.a.a

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 131 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $2 + 129\cdot 131 + 3\cdot 131^{2} + 118\cdot 131^{3} + 70\cdot 131^{4} +O\left(131^{ 5 }\right)$ $r_{ 2 }$ $=$ $16 + 112\cdot 131 + 72\cdot 131^{2} + 124\cdot 131^{3} + 36\cdot 131^{4} +O\left(131^{ 5 }\right)$ $r_{ 3 }$ $=$ $43 + 14\cdot 131 + 37\cdot 131^{2} + 104\cdot 131^{3} + 5\cdot 131^{4} +O\left(131^{ 5 }\right)$ $r_{ 4 }$ $=$ $68 + 75\cdot 131 + 38\cdot 131^{2} + 30\cdot 131^{3} + 25\cdot 131^{4} +O\left(131^{ 5 }\right)$ $r_{ 5 }$ $=$ $72 + 104\cdot 131 + 34\cdot 131^{2} + 104\cdot 131^{3} + 68\cdot 131^{4} +O\left(131^{ 5 }\right)$ $r_{ 6 }$ $=$ $86 + 16\cdot 131 + 87\cdot 131^{2} + 94\cdot 131^{3} + 60\cdot 131^{4} +O\left(131^{ 5 }\right)$ $r_{ 7 }$ $=$ $114 + 85\cdot 131 + 91\cdot 131^{2} + 84\cdot 131^{3} + 14\cdot 131^{4} +O\left(131^{ 5 }\right)$ $r_{ 8 }$ $=$ $124 + 116\cdot 131 + 26\cdot 131^{2} + 125\cdot 131^{3} + 109\cdot 131^{4} +O\left(131^{ 5 }\right)$

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

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

Character values on conjugacy classes

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