Properties

Label 2.3e2_7e2_11e2.8t5.1c1
Dimension 2
Group $Q_8$
Conductor $ 3^{2} \cdot 7^{2} \cdot 11^{2}$
Root number -1
Frobenius-Schur indicator -1

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Basic invariants

Dimension:$2$
Group:$Q_8$
Conductor:$53361= 3^{2} \cdot 7^{2} \cdot 11^{2} $
Artin number field: Splitting field of $f= x^{8} - 3 x^{7} - 62 x^{6} + 66 x^{5} + 1125 x^{4} + 264 x^{3} - 4982 x^{2} - 4245 x + 823 $ 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 $\Q_{ 167 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 6 + 140\cdot 167 + 18\cdot 167^{2} + 39\cdot 167^{3} + 151\cdot 167^{4} +O\left(167^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 9 + 11\cdot 167 + 160\cdot 167^{2} + 89\cdot 167^{3} + 105\cdot 167^{4} +O\left(167^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 22 + 119\cdot 167 + 72\cdot 167^{2} + 159\cdot 167^{3} + 117\cdot 167^{4} +O\left(167^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 29 + 159\cdot 167 + 119\cdot 167^{2} + 95\cdot 167^{3} + 154\cdot 167^{4} +O\left(167^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 31 + 4\cdot 167 + 82\cdot 167^{2} + 40\cdot 167^{3} + 164\cdot 167^{4} +O\left(167^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 88 + 14\cdot 167 + 73\cdot 167^{2} + 95\cdot 167^{3} + 119\cdot 167^{4} +O\left(167^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 154 + 152\cdot 167 + 91\cdot 167^{2} + 12\cdot 167^{3} + 84\cdot 167^{4} +O\left(167^{ 5 }\right)$
$r_{ 8 }$ $=$ $ 165 + 66\cdot 167 + 49\cdot 167^{2} + 135\cdot 167^{3} + 104\cdot 167^{4} +O\left(167^{ 5 }\right)$

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

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

Character values on conjugacy classes

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