Properties

Label 2.2004.8t6.b
Dimension $2$
Group $D_{8}$
Conductor $2004$
Indicator $1$

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Basic invariants

Dimension:$2$
Group:$D_{8}$
Conductor:\(2004\)\(\medspace = 2^{2} \cdot 3 \cdot 167 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 8.2.96577152768.2
Galois orbit size: $2$
Smallest permutation container: $D_{8}$
Parity: odd
Projective image: $D_4$
Projective field: Galois closure of 4.2.24048.2

Galois action

Roots of defining polynomial

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

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 values
$c1$ $c2$
$1$ $1$ $()$ $2$ $2$
$1$ $2$ $(1,2)(3,5)(4,8)(6,7)$ $-2$ $-2$
$4$ $2$ $(1,2)(4,7)(6,8)$ $0$ $0$
$4$ $2$ $(1,7)(2,6)(3,8)(4,5)$ $0$ $0$
$2$ $4$ $(1,3,2,5)(4,6,8,7)$ $0$ $0$
$2$ $8$ $(1,6,3,8,2,7,5,4)$ $-\zeta_{8}^{3} + \zeta_{8}$ $\zeta_{8}^{3} - \zeta_{8}$
$2$ $8$ $(1,8,5,6,2,4,3,7)$ $\zeta_{8}^{3} - \zeta_{8}$ $-\zeta_{8}^{3} + \zeta_{8}$
The blue line marks the conjugacy class containing complex conjugation.