Properties

Label 2.2e8_17e2.8t5.1c1
Dimension 2
Group $Q_8$
Conductor $ 2^{8} \cdot 17^{2}$
Root number -1
Frobenius-Schur indicator -1

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Basic invariants

Dimension:$2$
Group:$Q_8$
Conductor:$73984= 2^{8} \cdot 17^{2} $
Artin number field: Splitting field of $f= x^{8} - 68 x^{6} + 986 x^{4} - 4624 x^{2} + 4624 $ 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 $\Q_{ 137 }$ to precision 10.
Roots:
$r_{ 1 }$ $=$ $ 2 + 15\cdot 137 + 21\cdot 137^{2} + 110\cdot 137^{3} + 90\cdot 137^{4} + 107\cdot 137^{5} + 129\cdot 137^{6} + 25\cdot 137^{7} + 64\cdot 137^{8} + 103\cdot 137^{9} +O\left(137^{ 10 }\right)$
$r_{ 2 }$ $=$ $ 3 + 71\cdot 137 + 26\cdot 137^{2} + 137^{3} + 82\cdot 137^{4} + 23\cdot 137^{5} + 17\cdot 137^{6} + 50\cdot 137^{7} + 131\cdot 137^{8} + 37\cdot 137^{9} +O\left(137^{ 10 }\right)$
$r_{ 3 }$ $=$ $ 47 + 121\cdot 137 + 22\cdot 137^{2} + 28\cdot 137^{3} + 44\cdot 137^{4} + 95\cdot 137^{5} + 66\cdot 137^{6} + 103\cdot 137^{7} + 55\cdot 137^{8} + 65\cdot 137^{9} +O\left(137^{ 10 }\right)$
$r_{ 4 }$ $=$ $ 60 + 136\cdot 137 + 94\cdot 137^{2} + 99\cdot 137^{3} + 84\cdot 137^{4} + 63\cdot 137^{5} + 52\cdot 137^{6} + 70\cdot 137^{7} + 50\cdot 137^{8} + 33\cdot 137^{9} +O\left(137^{ 10 }\right)$
$r_{ 5 }$ $=$ $ 77 + 42\cdot 137^{2} + 37\cdot 137^{3} + 52\cdot 137^{4} + 73\cdot 137^{5} + 84\cdot 137^{6} + 66\cdot 137^{7} + 86\cdot 137^{8} + 103\cdot 137^{9} +O\left(137^{ 10 }\right)$
$r_{ 6 }$ $=$ $ 90 + 15\cdot 137 + 114\cdot 137^{2} + 108\cdot 137^{3} + 92\cdot 137^{4} + 41\cdot 137^{5} + 70\cdot 137^{6} + 33\cdot 137^{7} + 81\cdot 137^{8} + 71\cdot 137^{9} +O\left(137^{ 10 }\right)$
$r_{ 7 }$ $=$ $ 134 + 65\cdot 137 + 110\cdot 137^{2} + 135\cdot 137^{3} + 54\cdot 137^{4} + 113\cdot 137^{5} + 119\cdot 137^{6} + 86\cdot 137^{7} + 5\cdot 137^{8} + 99\cdot 137^{9} +O\left(137^{ 10 }\right)$
$r_{ 8 }$ $=$ $ 135 + 121\cdot 137 + 115\cdot 137^{2} + 26\cdot 137^{3} + 46\cdot 137^{4} + 29\cdot 137^{5} + 7\cdot 137^{6} + 111\cdot 137^{7} + 72\cdot 137^{8} + 33\cdot 137^{9} +O\left(137^{ 10 }\right)$

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

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

Character values on conjugacy classes

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