Properties

Label 2.3e2_5e2_29e2.8t5.1c1
Dimension 2
Group $Q_8$
Conductor $ 3^{2} \cdot 5^{2} \cdot 29^{2}$
Root number 1
Frobenius-Schur indicator -1

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Basic invariants

Dimension:$2$
Group:$Q_8$
Conductor:$189225= 3^{2} \cdot 5^{2} \cdot 29^{2} $
Artin number field: Splitting field of $f= x^{8} - x^{7} + 98 x^{6} - 105 x^{5} + 3191 x^{4} + 1665 x^{3} + 44072 x^{2} + 47933 x + 328171 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $Q_8$
Parity: Even
Determinant: 1.1.1t1.1c1

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\left(109^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 7 + 89\cdot 109 + 51\cdot 109^{2} + 53\cdot 109^{3} + 57\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 27 + 62\cdot 109 + 77\cdot 109^{2} + 45\cdot 109^{3} + 53\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 47 + 17\cdot 109 + 10\cdot 109^{2} + 100\cdot 109^{3} + 106\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 63 + 21\cdot 109 + 20\cdot 109^{2} + 31\cdot 109^{3} + 37\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 92 + 90\cdot 109 + 103\cdot 109^{2} + 82\cdot 109^{3} + 101\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 94 + 20\cdot 109 + 66\cdot 109^{2} + 94\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 8 }$ $=$ $ 102 + 12\cdot 109 + 31\cdot 109^{2} + 40\cdot 109^{3} + 7\cdot 109^{4} +O\left(109^{ 5 }\right)$

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 value
$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.