Properties

Label 2.2e2_3_167.8t6.2c2
Dimension 2
Group $D_{8}$
Conductor $ 2^{2} \cdot 3 \cdot 167 $
Root number 1
Frobenius-Schur indicator 1

Related objects

Learn more about

Basic invariants

Dimension:$2$
Group:$D_{8}$
Conductor:$2004= 2^{2} \cdot 3 \cdot 167 $
Artin number field: Splitting field of $f= x^{8} - 11 x^{6} + 64 x^{4} - 60 x^{3} - 15 x^{2} - 36 x + 9 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $D_{8}$
Parity: Odd
Determinant: 1.2e2_3_167.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 179 }$ to precision 6.
Roots:
$r_{ 1 }$ $=$ $ 6 + 112\cdot 179 + 43\cdot 179^{3} + 106\cdot 179^{4} + 17\cdot 179^{5} +O\left(179^{ 6 }\right)$
$r_{ 2 }$ $=$ $ 23 + 40\cdot 179 + 85\cdot 179^{2} + 30\cdot 179^{3} + 29\cdot 179^{4} + 110\cdot 179^{5} +O\left(179^{ 6 }\right)$
$r_{ 3 }$ $=$ $ 35 + 131\cdot 179 + 21\cdot 179^{2} + 119\cdot 179^{3} + 66\cdot 179^{4} + 122\cdot 179^{5} +O\left(179^{ 6 }\right)$
$r_{ 4 }$ $=$ $ 86 + 47\cdot 179 + 67\cdot 179^{2} + 26\cdot 179^{3} + 131\cdot 179^{4} + 34\cdot 179^{5} +O\left(179^{ 6 }\right)$
$r_{ 5 }$ $=$ $ 96 + 140\cdot 179 + 157\cdot 179^{2} + 153\cdot 179^{3} + 93\cdot 179^{4} + 140\cdot 179^{5} +O\left(179^{ 6 }\right)$
$r_{ 6 }$ $=$ $ 135 + 8\cdot 179 + 158\cdot 179^{2} + 164\cdot 179^{3} + 108\cdot 179^{4} + 111\cdot 179^{5} +O\left(179^{ 6 }\right)$
$r_{ 7 }$ $=$ $ 166 + 154\cdot 179 + 163\cdot 179^{2} + 125\cdot 179^{3} + 70\cdot 179^{4} + 116\cdot 179^{5} +O\left(179^{ 6 }\right)$
$r_{ 8 }$ $=$ $ 169 + 80\cdot 179 + 61\cdot 179^{2} + 52\cdot 179^{3} + 109\cdot 179^{4} + 62\cdot 179^{5} +O\left(179^{ 6 }\right)$

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

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

Character values on conjugacy classes

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