Properties

Label 2.273.8t17.d
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 8.0.8719893.2
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 $\Q_{ 103 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ \( 7 + 97\cdot 103 + 86\cdot 103^{2} + 66\cdot 103^{3} + 43\cdot 103^{4} +O(103^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 10 + 51\cdot 103 + 32\cdot 103^{2} + 35\cdot 103^{3} + 79\cdot 103^{4} +O(103^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 18 + 34\cdot 103 + 62\cdot 103^{2} + 66\cdot 103^{3} + 55\cdot 103^{4} +O(103^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 19 + 51\cdot 103 + 83\cdot 103^{2} + 89\cdot 103^{3} + 43\cdot 103^{4} +O(103^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 41 + 98\cdot 103 + 68\cdot 103^{2} + 28\cdot 103^{3} + 100\cdot 103^{4} +O(103^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 44 + 2\cdot 103 + 14\cdot 103^{2} + 22\cdot 103^{3} + 6\cdot 103^{4} +O(103^{5})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 81 + 38\cdot 103 + 102\cdot 103^{2} + 80\cdot 103^{3} + 17\cdot 103^{4} +O(103^{5})\)  Toggle raw display
$r_{ 8 }$ $=$ \( 92 + 38\cdot 103 + 64\cdot 103^{2} + 21\cdot 103^{3} + 65\cdot 103^{4} +O(103^{5})\)  Toggle raw display

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

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

Character values on conjugacy classes

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