Properties

Label 2.1520.8t6.b
Dimension $2$
Group $D_{8}$
Conductor $1520$
Indicator $1$

Related objects

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Basic invariants

Dimension:$2$
Group:$D_{8}$
Conductor:\(1520\)\(\medspace = 2^{4} \cdot 5 \cdot 19 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 8.2.1097440000.5
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:
$r_{ 1 }$ $=$ \( 13 + 22\cdot 191 + 117\cdot 191^{2} + 125\cdot 191^{3} + 136\cdot 191^{4} + 83\cdot 191^{5} +O(191^{6})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 30 + 40\cdot 191 + 171\cdot 191^{2} + 106\cdot 191^{3} + 116\cdot 191^{4} + 158\cdot 191^{5} +O(191^{6})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 37 + 26\cdot 191 + 11\cdot 191^{2} + 96\cdot 191^{3} + 144\cdot 191^{4} + 104\cdot 191^{5} +O(191^{6})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 68 + 114\cdot 191 + 45\cdot 191^{2} + 159\cdot 191^{3} + 13\cdot 191^{4} + 77\cdot 191^{5} +O(191^{6})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 123 + 76\cdot 191 + 145\cdot 191^{2} + 31\cdot 191^{3} + 177\cdot 191^{4} + 113\cdot 191^{5} +O(191^{6})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 154 + 164\cdot 191 + 179\cdot 191^{2} + 94\cdot 191^{3} + 46\cdot 191^{4} + 86\cdot 191^{5} +O(191^{6})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 161 + 150\cdot 191 + 19\cdot 191^{2} + 84\cdot 191^{3} + 74\cdot 191^{4} + 32\cdot 191^{5} +O(191^{6})\)  Toggle raw display
$r_{ 8 }$ $=$ \( 178 + 168\cdot 191 + 73\cdot 191^{2} + 65\cdot 191^{3} + 54\cdot 191^{4} + 107\cdot 191^{5} +O(191^{6})\)  Toggle raw display

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

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

Character values on conjugacy classes

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