Properties

Label 2.273.8t17.c
Dimension $2$
Group $C_4\wr C_2$
Conductor $273$
Indicator $0$

Related objects

Learn more

Basic invariants

Dimension:$2$
Group:$C_4\wr C_2$
Conductor:\(273\)\(\medspace = 3 \cdot 7 \cdot 13 \)
Artin number field: Galois closure of 8.0.8719893.1
Galois orbit size: $2$
Smallest permutation container: $C_4\wr C_2$
Parity: odd
Projective image: D_4
Projective field: 4.2.322959.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 181 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ \( 3 + 22\cdot 181 + 119\cdot 181^{2} + 127\cdot 181^{3} + 146\cdot 181^{4} +O(181^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 10 + 109\cdot 181 + 174\cdot 181^{2} + 179\cdot 181^{3} + 64\cdot 181^{4} +O(181^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 36 + 155\cdot 181 + 70\cdot 181^{2} + 151\cdot 181^{3} + 124\cdot 181^{4} +O(181^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 42 + 145\cdot 181 + 181^{2} + 36\cdot 181^{3} + 118\cdot 181^{4} +O(181^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 67 + 89\cdot 181 + 109\cdot 181^{2} + 58\cdot 181^{3} + 121\cdot 181^{4} +O(181^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 119 + 103\cdot 181 + 47\cdot 181^{2} + 136\cdot 181^{3} + 85\cdot 181^{4} +O(181^{5})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 127 + 82\cdot 181 + 52\cdot 181^{2} + 106\cdot 181^{3} + 157\cdot 181^{4} +O(181^{5})\)  Toggle raw display
$r_{ 8 }$ $=$ \( 142 + 16\cdot 181 + 148\cdot 181^{2} + 108\cdot 181^{3} + 85\cdot 181^{4} +O(181^{5})\)  Toggle raw display

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

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

Character values on conjugacy classes

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