Properties

Label 2.2e4_3e2_5e2_7e2.8t5.1c1
Dimension 2
Group $Q_8$
Conductor $ 2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$
Root number -1
Frobenius-Schur indicator -1

Related objects

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Basic invariants

Dimension:$2$
Group:$Q_8$
Conductor:$176400= 2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} $
Artin number field: Splitting field of $f= x^{8} - 105 x^{6} + 3465 x^{4} - 44100 x^{2} + 176400 $ 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 11.
Roots:
$r_{ 1 }$ $=$ $ 17 + 11\cdot 109 + 47\cdot 109^{2} + 81\cdot 109^{3} + 11\cdot 109^{4} + 96\cdot 109^{5} + 63\cdot 109^{6} + 91\cdot 109^{7} + 6\cdot 109^{8} + 87\cdot 109^{9} + 13\cdot 109^{10} +O\left(109^{ 11 }\right)$
$r_{ 2 }$ $=$ $ 27 + 17\cdot 109 + 67\cdot 109^{2} + 39\cdot 109^{3} + 40\cdot 109^{4} + 85\cdot 109^{5} + 9\cdot 109^{6} + 63\cdot 109^{7} + 99\cdot 109^{8} + 29\cdot 109^{9} + 77\cdot 109^{10} +O\left(109^{ 11 }\right)$
$r_{ 3 }$ $=$ $ 31 + 101\cdot 109 + 91\cdot 109^{2} + 8\cdot 109^{3} + 98\cdot 109^{4} + 35\cdot 109^{5} + 34\cdot 109^{6} + 91\cdot 109^{7} + 46\cdot 109^{8} + 52\cdot 109^{9} + 83\cdot 109^{10} +O\left(109^{ 11 }\right)$
$r_{ 4 }$ $=$ $ 52 + 15\cdot 109 + 61\cdot 109^{2} + 7\cdot 109^{3} + 29\cdot 109^{4} + 27\cdot 109^{5} + 18\cdot 109^{6} + 78\cdot 109^{7} + 19\cdot 109^{8} + 34\cdot 109^{9} + 9\cdot 109^{10} +O\left(109^{ 11 }\right)$
$r_{ 5 }$ $=$ $ 57 + 93\cdot 109 + 47\cdot 109^{2} + 101\cdot 109^{3} + 79\cdot 109^{4} + 81\cdot 109^{5} + 90\cdot 109^{6} + 30\cdot 109^{7} + 89\cdot 109^{8} + 74\cdot 109^{9} + 99\cdot 109^{10} +O\left(109^{ 11 }\right)$
$r_{ 6 }$ $=$ $ 78 + 7\cdot 109 + 17\cdot 109^{2} + 100\cdot 109^{3} + 10\cdot 109^{4} + 73\cdot 109^{5} + 74\cdot 109^{6} + 17\cdot 109^{7} + 62\cdot 109^{8} + 56\cdot 109^{9} + 25\cdot 109^{10} +O\left(109^{ 11 }\right)$
$r_{ 7 }$ $=$ $ 82 + 91\cdot 109 + 41\cdot 109^{2} + 69\cdot 109^{3} + 68\cdot 109^{4} + 23\cdot 109^{5} + 99\cdot 109^{6} + 45\cdot 109^{7} + 9\cdot 109^{8} + 79\cdot 109^{9} + 31\cdot 109^{10} +O\left(109^{ 11 }\right)$
$r_{ 8 }$ $=$ $ 92 + 97\cdot 109 + 61\cdot 109^{2} + 27\cdot 109^{3} + 97\cdot 109^{4} + 12\cdot 109^{5} + 45\cdot 109^{6} + 17\cdot 109^{7} + 102\cdot 109^{8} + 21\cdot 109^{9} + 95\cdot 109^{10} +O\left(109^{ 11 }\right)$

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

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

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