Properties

Label 2.5_19.8t6.1c2
Dimension 2
Group $D_{8}$
Conductor $ 5 \cdot 19 $
Root number 1
Frobenius-Schur indicator 1

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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: 1.5_19.2t1.1c1

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

SizeOrderAction 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.