Properties

Label 2.2e8_3e2_5e2.8t5.1c1
Dimension 2
Group $Q_8$
Conductor $ 2^{8} \cdot 3^{2} \cdot 5^{2}$
Root number -1
Frobenius-Schur indicator -1

Related objects

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Basic invariants

Dimension:$2$
Group:$Q_8$
Conductor:$57600= 2^{8} \cdot 3^{2} \cdot 5^{2} $
Artin number field: Splitting field of $f= x^{8} - 60 x^{6} + 810 x^{4} - 1800 x^{2} + 900 $ 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 $\Q_{ 149 }$ to precision 10.
Roots:
$r_{ 1 }$ $=$ $ 1 + 140\cdot 149 + 95\cdot 149^{2} + 72\cdot 149^{3} + 116\cdot 149^{4} + 77\cdot 149^{5} + 91\cdot 149^{6} + 45\cdot 149^{7} + 38\cdot 149^{8} + 148\cdot 149^{9} +O\left(149^{ 10 }\right)$
$r_{ 2 }$ $=$ $ 40 + 28\cdot 149 + 78\cdot 149^{2} + 90\cdot 149^{3} + 40\cdot 149^{4} + 44\cdot 149^{5} + 101\cdot 149^{6} + 135\cdot 149^{7} + 133\cdot 149^{8} + 82\cdot 149^{9} +O\left(149^{ 10 }\right)$
$r_{ 3 }$ $=$ $ 46 + 145\cdot 149 + 22\cdot 149^{2} + 90\cdot 149^{3} + 70\cdot 149^{4} + 47\cdot 149^{5} + 69\cdot 149^{6} + 89\cdot 149^{7} + 46\cdot 149^{8} + 43\cdot 149^{9} +O\left(149^{ 10 }\right)$
$r_{ 4 }$ $=$ $ 51 + 34\cdot 149 + 96\cdot 149^{2} + 89\cdot 149^{3} + 31\cdot 149^{4} + 110\cdot 149^{5} + 34\cdot 149^{6} + 59\cdot 149^{7} + 147\cdot 149^{8} + 21\cdot 149^{9} +O\left(149^{ 10 }\right)$
$r_{ 5 }$ $=$ $ 98 + 114\cdot 149 + 52\cdot 149^{2} + 59\cdot 149^{3} + 117\cdot 149^{4} + 38\cdot 149^{5} + 114\cdot 149^{6} + 89\cdot 149^{7} + 149^{8} + 127\cdot 149^{9} +O\left(149^{ 10 }\right)$
$r_{ 6 }$ $=$ $ 103 + 3\cdot 149 + 126\cdot 149^{2} + 58\cdot 149^{3} + 78\cdot 149^{4} + 101\cdot 149^{5} + 79\cdot 149^{6} + 59\cdot 149^{7} + 102\cdot 149^{8} + 105\cdot 149^{9} +O\left(149^{ 10 }\right)$
$r_{ 7 }$ $=$ $ 109 + 120\cdot 149 + 70\cdot 149^{2} + 58\cdot 149^{3} + 108\cdot 149^{4} + 104\cdot 149^{5} + 47\cdot 149^{6} + 13\cdot 149^{7} + 15\cdot 149^{8} + 66\cdot 149^{9} +O\left(149^{ 10 }\right)$
$r_{ 8 }$ $=$ $ 148 + 8\cdot 149 + 53\cdot 149^{2} + 76\cdot 149^{3} + 32\cdot 149^{4} + 71\cdot 149^{5} + 57\cdot 149^{6} + 103\cdot 149^{7} + 110\cdot 149^{8} +O\left(149^{ 10 }\right)$

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

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

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$$4$$(1,2,8,7)(3,5,6,4)$$0$
$2$$4$$(1,4,8,5)(2,6,7,3)$$0$
$2$$4$$(1,6,8,3)(2,5,7,4)$$0$
The blue line marks the conjugacy class containing complex conjugation.