Properties

Label 2.855.8t6.b
Dimension $2$
Group $D_{8}$
Conductor $855$
Indicator $1$

Related objects

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Basic invariants

Dimension:$2$
Group:$D_{8}$
Conductor:\(855\)\(\medspace = 3^{2} \cdot 5 \cdot 19 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 8.2.347236875.1
Galois orbit size: $2$
Smallest permutation container: $D_{8}$
Parity: odd
Projective image: $D_4$
Projective field: 4.2.475.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 191 }$ to precision 6.
Roots:
$r_{ 1 }$ $=$ \( 6 + 139\cdot 191 + 188\cdot 191^{2} + 94\cdot 191^{3} + 27\cdot 191^{4} + 156\cdot 191^{5} +O(191^{6})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 49 + 116\cdot 191 + 21\cdot 191^{2} + 34\cdot 191^{3} + 99\cdot 191^{4} + 24\cdot 191^{5} +O(191^{6})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 73 + 145\cdot 191 + 93\cdot 191^{2} + 161\cdot 191^{3} + 179\cdot 191^{4} + 115\cdot 191^{5} +O(191^{6})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 78 + 85\cdot 191 + 157\cdot 191^{2} + 140\cdot 191^{3} + 21\cdot 191^{4} + 144\cdot 191^{5} +O(191^{6})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 108 + 135\cdot 191 + 118\cdot 191^{2} + 172\cdot 191^{3} + 17\cdot 191^{4} + 182\cdot 191^{5} +O(191^{6})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 122 + 54\cdot 191 + 80\cdot 191^{2} + 66\cdot 191^{3} + 150\cdot 191^{4} + 84\cdot 191^{5} +O(191^{6})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 153 + 134\cdot 191 + 164\cdot 191^{2} + 7\cdot 191^{3} + 95\cdot 191^{4} + 44\cdot 191^{5} +O(191^{6})\)  Toggle raw display
$r_{ 8 }$ $=$ \( 178 + 143\cdot 191 + 129\cdot 191^{2} + 85\cdot 191^{3} + 172\cdot 191^{4} + 11\cdot 191^{5} +O(191^{6})\)  Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 8 }$ Character values
$c1$ $c2$
$1$ $1$ $()$ $2$ $2$
$1$ $2$ $(1,3)(2,7)(4,6)(5,8)$ $-2$ $-2$
$4$ $2$ $(1,6)(2,5)(3,4)(7,8)$ $0$ $0$
$4$ $2$ $(2,7)(4,8)(5,6)$ $0$ $0$
$2$ $4$ $(1,7,3,2)(4,8,6,5)$ $0$ $0$
$2$ $8$ $(1,5,7,4,3,8,2,6)$ $-\zeta_{8}^{3} + \zeta_{8}$ $\zeta_{8}^{3} - \zeta_{8}$
$2$ $8$ $(1,4,2,5,3,6,7,8)$ $\zeta_{8}^{3} - \zeta_{8}$ $-\zeta_{8}^{3} + \zeta_{8}$
The blue line marks the conjugacy class containing complex conjugation.