Properties

Label 2.189225.8t5.a
Dimension $2$
Group $Q_8$
Conductor $189225$
Indicator $-1$

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Basic invariants

Dimension:$2$
Group:$Q_8$
Conductor:\(189225\)\(\medspace = 3^{2} \cdot 5^{2} \cdot 29^{2} \)
Frobenius-Schur indicator: $-1$
Root number: $1$
Artin number field: Galois closure of 8.0.752823265640625.1
Galois orbit size: $1$
Smallest permutation container: $Q_8$
Parity: even
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(\sqrt{5}, \sqrt{29})\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 109 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ \( 5 + 12\cdot 109 + 75\cdot 109^{2} + 81\cdot 109^{3} + 86\cdot 109^{4} +O(109^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 7 + 89\cdot 109 + 51\cdot 109^{2} + 53\cdot 109^{3} + 57\cdot 109^{4} +O(109^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 27 + 62\cdot 109 + 77\cdot 109^{2} + 45\cdot 109^{3} + 53\cdot 109^{4} +O(109^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 47 + 17\cdot 109 + 10\cdot 109^{2} + 100\cdot 109^{3} + 106\cdot 109^{4} +O(109^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 63 + 21\cdot 109 + 20\cdot 109^{2} + 31\cdot 109^{3} + 37\cdot 109^{4} +O(109^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 92 + 90\cdot 109 + 103\cdot 109^{2} + 82\cdot 109^{3} + 101\cdot 109^{4} +O(109^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 94 + 20\cdot 109 + 66\cdot 109^{2} + 94\cdot 109^{4} +O(109^{5})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 102 + 12\cdot 109 + 31\cdot 109^{2} + 40\cdot 109^{3} + 7\cdot 109^{4} +O(109^{5})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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