Properties

Label 2.1156.8t17.a
Dimension $2$
Group $C_4\wr C_2$
Conductor $1156$
Indicator $0$

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Basic invariants

Dimension:$2$
Group:$C_4\wr C_2$
Conductor:\(1156\)\(\medspace = 2^{2} \cdot 17^{2}\)
Artin number field: Galois closure of 8.0.1257728.1
Galois orbit size: $2$
Smallest permutation container: $C_4\wr C_2$
Parity: odd
Projective image: $D_4$
Projective field: 4.2.19652.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 149 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ \( 23 + 19\cdot 149 + 46\cdot 149^{2} + 81\cdot 149^{3} + 103\cdot 149^{4} +O(149^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 61 + 34\cdot 149 + 129\cdot 149^{2} + 79\cdot 149^{3} + 110\cdot 149^{4} +O(149^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 81 + 8\cdot 149 + 91\cdot 149^{2} + 83\cdot 149^{3} + 71\cdot 149^{4} +O(149^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 98 + 18\cdot 149 + 67\cdot 149^{3} + 149^{4} +O(149^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 100 + 63\cdot 149 + 100\cdot 149^{2} + 70\cdot 149^{3} + 78\cdot 149^{4} +O(149^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 122 + 35\cdot 149 + 36\cdot 149^{2} + 22\cdot 149^{3} + 101\cdot 149^{4} +O(149^{5})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 124 + 114\cdot 149 + 49\cdot 149^{2} + 142\cdot 149^{3} + 36\cdot 149^{4} +O(149^{5})\)  Toggle raw display
$r_{ 8 }$ $=$ \( 138 + 2\cdot 149 + 143\cdot 149^{2} + 48\cdot 149^{3} + 92\cdot 149^{4} +O(149^{5})\)  Toggle raw display

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

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

Character values on conjugacy classes

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