Properties

Label 2.3_37.8t6.1c1
Dimension 2
Group $D_{8}$
Conductor $ 3 \cdot 37 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$2$
Group:$D_{8}$
Conductor:$111= 3 \cdot 37 $
Artin number field: Splitting field of $f= x^{8} - 3 x^{7} + 3 x^{6} - 3 x^{5} + 5 x^{4} - 6 x^{3} + 6 x^{2} - 3 x + 1 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $D_{8}$
Parity: Odd
Determinant: 1.3_37.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 127 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 32 + 15\cdot 127 + 60\cdot 127^{2} + 78\cdot 127^{3} + 55\cdot 127^{4} +O\left(127^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 51 + 34\cdot 127 + 51\cdot 127^{2} + 96\cdot 127^{3} + 108\cdot 127^{4} +O\left(127^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 64 + 34\cdot 127 + 104\cdot 127^{2} + 8\cdot 127^{3} + 65\cdot 127^{4} +O\left(127^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 89 + 3\cdot 127 + 48\cdot 127^{2} + 60\cdot 127^{3} + 53\cdot 127^{4} +O\left(127^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 93 + 72\cdot 127 + 58\cdot 127^{2} + 57\cdot 127^{3} + 105\cdot 127^{4} +O\left(127^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 94 + 22\cdot 127 + 57\cdot 127^{2} + 23\cdot 127^{3} + 60\cdot 127^{4} +O\left(127^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 104 + 113\cdot 127 + 85\cdot 127^{2} + 100\cdot 127^{3} + 109\cdot 127^{4} +O\left(127^{ 5 }\right)$
$r_{ 8 }$ $=$ $ 111 + 83\cdot 127 + 42\cdot 127^{2} + 82\cdot 127^{3} + 76\cdot 127^{4} +O\left(127^{ 5 }\right)$

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

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

Character values on conjugacy classes

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