Properties

Label 2.1280.8t17.a
Dimension $2$
Group $C_4\wr C_2$
Conductor $1280$
Indicator $0$

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Basic invariants

Dimension:$2$
Group:$C_4\wr C_2$
Conductor:\(1280\)\(\medspace = 2^{8} \cdot 5 \)
Artin number field: Galois closure of 8.0.2097152000.3
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 $\Q_{ 109 }$ to precision 7.
Roots:
$r_{ 1 }$ $=$ \( 11 + 11\cdot 109 + 77\cdot 109^{3} + 19\cdot 109^{4} + 18\cdot 109^{5} + 84\cdot 109^{6} +O(109^{7})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 25 + 108\cdot 109 + 92\cdot 109^{2} + 2\cdot 109^{3} + 41\cdot 109^{4} + 21\cdot 109^{5} + 48\cdot 109^{6} +O(109^{7})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 36 + 109 + 24\cdot 109^{2} + 77\cdot 109^{3} + 58\cdot 109^{4} + 6\cdot 109^{5} + 75\cdot 109^{6} +O(109^{7})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 47 + 42\cdot 109 + 74\cdot 109^{2} + 39\cdot 109^{3} + 89\cdot 109^{4} + 74\cdot 109^{5} + 53\cdot 109^{6} +O(109^{7})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 62 + 66\cdot 109 + 34\cdot 109^{2} + 69\cdot 109^{3} + 19\cdot 109^{4} + 34\cdot 109^{5} + 55\cdot 109^{6} +O(109^{7})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 73 + 107\cdot 109 + 84\cdot 109^{2} + 31\cdot 109^{3} + 50\cdot 109^{4} + 102\cdot 109^{5} + 33\cdot 109^{6} +O(109^{7})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 84 + 16\cdot 109^{2} + 106\cdot 109^{3} + 67\cdot 109^{4} + 87\cdot 109^{5} + 60\cdot 109^{6} +O(109^{7})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 98 + 97\cdot 109 + 108\cdot 109^{2} + 31\cdot 109^{3} + 89\cdot 109^{4} + 90\cdot 109^{5} + 24\cdot 109^{6} +O(109^{7})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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