Properties

Label 2.1216.8t11.a
Dimension $2$
Group $Q_8:C_2$
Conductor $1216$
Indicator $0$

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Basic invariants

Dimension:$2$
Group:$Q_8:C_2$
Conductor:\(1216\)\(\medspace = 2^{6} \cdot 19 \)
Artin number field: Galois closure of 8.0.378535936.1
Galois orbit size: $2$
Smallest permutation container: $Q_8:C_2$
Parity: odd
Projective image: $C_2^2$
Projective field: \(\Q(\sqrt{-2}, \sqrt{-19})\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 137 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ \( 30 + 54\cdot 137 + 7\cdot 137^{2} + 122\cdot 137^{3} + 33\cdot 137^{4} +O(137^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 55 + 86\cdot 137 + 97\cdot 137^{2} + 55\cdot 137^{3} + 35\cdot 137^{4} +O(137^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 71 + 34\cdot 137 + 42\cdot 137^{2} + 87\cdot 137^{3} + 72\cdot 137^{4} +O(137^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 86 + 3\cdot 137 + 105\cdot 137^{2} + 126\cdot 137^{3} + 73\cdot 137^{4} +O(137^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 89 + 44\cdot 137 + 119\cdot 137^{2} + 74\cdot 137^{3} + 93\cdot 137^{4} +O(137^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 103 + 27\cdot 137 + 87\cdot 137^{2} + 77\cdot 137^{3} + 114\cdot 137^{4} +O(137^{5})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 120 + 98\cdot 137 + 126\cdot 137^{2} + 8\cdot 137^{3} + 132\cdot 137^{4} +O(137^{5})\)  Toggle raw display
$r_{ 8 }$ $=$ \( 135 + 60\cdot 137 + 99\cdot 137^{2} + 131\cdot 137^{3} + 128\cdot 137^{4} +O(137^{5})\)  Toggle raw display

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

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

Character values on conjugacy classes

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