Properties

Label 2.3e2_7e2_17e2.8t5.2c1
Dimension 2
Group $Q_8$
Conductor $ 3^{2} \cdot 7^{2} \cdot 17^{2}$
Root number 1
Frobenius-Schur indicator -1

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Basic invariants

Dimension:$2$
Group:$Q_8$
Conductor:$127449= 3^{2} \cdot 7^{2} \cdot 17^{2} $
Artin number field: Splitting field of $f= x^{8} - x^{7} + 65 x^{6} - 439 x^{5} + 1876 x^{4} - 12191 x^{3} + 60887 x^{2} - 124718 x + 121291 $ 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 $\Q_{ 89 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 17 + 59\cdot 89 + 69\cdot 89^{2} + 71\cdot 89^{3} + 27\cdot 89^{4} +O\left(89^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 19 + 75\cdot 89 + 57\cdot 89^{2} + 78\cdot 89^{3} + 11\cdot 89^{4} +O\left(89^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 24 + 65\cdot 89 + 57\cdot 89^{2} + 13\cdot 89^{3} + 44\cdot 89^{4} +O\left(89^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 29 + 66\cdot 89 + 82\cdot 89^{2} + 11\cdot 89^{3} + 10\cdot 89^{4} +O\left(89^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 33 + 24\cdot 89 + 11\cdot 89^{2} + 31\cdot 89^{3} + 12\cdot 89^{4} +O\left(89^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 67 + 82\cdot 89 + 53\cdot 89^{2} + 29\cdot 89^{3} + 20\cdot 89^{4} +O\left(89^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 82 + 66\cdot 89 + 31\cdot 89^{2} + 46\cdot 89^{3} + 19\cdot 89^{4} +O\left(89^{ 5 }\right)$
$r_{ 8 }$ $=$ $ 86 + 4\cdot 89 + 80\cdot 89^{2} + 72\cdot 89^{3} + 31\cdot 89^{4} +O\left(89^{ 5 }\right)$

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

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

Character values on conjugacy classes

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