Properties

Label 2.160.8t17.a
Dimension $2$
Group $C_4\wr C_2$
Conductor $160$
Indicator $0$

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Basic invariants

Dimension:$2$
Group:$C_4\wr C_2$
Conductor:\(160\)\(\medspace = 2^{5} \cdot 5 \)
Artin number field: Galois closure of 8.0.8192000.1
Galois orbit size: $2$
Smallest permutation container: $C_4\wr C_2$
Parity: odd
Projective image: $D_4$
Projective field: Galois closure of 4.2.2000.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 181 }$ to precision 6.
Roots:
$r_{ 1 }$ $=$ \( 13 + 27\cdot 181 + 151\cdot 181^{2} + 165\cdot 181^{3} + 106\cdot 181^{4} + 35\cdot 181^{5} +O(181^{6})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 62 + 22\cdot 181 + 172\cdot 181^{2} + 143\cdot 181^{3} + 44\cdot 181^{4} + 40\cdot 181^{5} +O(181^{6})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 66 + 37\cdot 181 + 46\cdot 181^{2} + 81\cdot 181^{3} + 159\cdot 181^{4} + 52\cdot 181^{5} +O(181^{6})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 89 + 26\cdot 181 + 149\cdot 181^{2} + 5\cdot 181^{3} + 90\cdot 181^{4} + 85\cdot 181^{5} +O(181^{6})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 92 + 154\cdot 181 + 31\cdot 181^{2} + 175\cdot 181^{3} + 90\cdot 181^{4} + 95\cdot 181^{5} +O(181^{6})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 115 + 143\cdot 181 + 134\cdot 181^{2} + 99\cdot 181^{3} + 21\cdot 181^{4} + 128\cdot 181^{5} +O(181^{6})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 119 + 158\cdot 181 + 8\cdot 181^{2} + 37\cdot 181^{3} + 136\cdot 181^{4} + 140\cdot 181^{5} +O(181^{6})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 168 + 153\cdot 181 + 29\cdot 181^{2} + 15\cdot 181^{3} + 74\cdot 181^{4} + 145\cdot 181^{5} +O(181^{6})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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