Properties

Label 5.2e6_743.6t16.1c1
Dimension 5
Group $S_6$
Conductor $ 2^{6} \cdot 743 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$5$
Group:$S_6$
Conductor:$47552= 2^{6} \cdot 743 $
Artin number field: Splitting field of $f= x^{6} - 2 x^{5} + 2 x^{4} - 2 x^{3} + x^{2} + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_6$
Parity: Odd
Determinant: 1.743.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 131 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 131 }$: $ x^{2} + 127 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 128 + 104\cdot 131 + 67\cdot 131^{2} + 47\cdot 131^{3} + 2\cdot 131^{4} +O\left(131^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 18 a + 12 + \left(101 a + 56\right)\cdot 131 + \left(67 a + 25\right)\cdot 131^{2} + \left(98 a + 127\right)\cdot 131^{3} + \left(63 a + 119\right)\cdot 131^{4} +O\left(131^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 113 a + 84 + \left(29 a + 49\right)\cdot 131 + \left(63 a + 64\right)\cdot 131^{2} + \left(32 a + 60\right)\cdot 131^{3} + \left(67 a + 14\right)\cdot 131^{4} +O\left(131^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 6 a + 81 + \left(57 a + 10\right)\cdot 131 + \left(6 a + 104\right)\cdot 131^{2} + \left(64 a + 97\right)\cdot 131^{3} + \left(12 a + 51\right)\cdot 131^{4} +O\left(131^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 116 + 69\cdot 131 + 58\cdot 131^{2} + 105\cdot 131^{3} + 35\cdot 131^{4} +O\left(131^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 125 a + 105 + \left(73 a + 101\right)\cdot 131 + \left(124 a + 72\right)\cdot 131^{2} + \left(66 a + 85\right)\cdot 131^{3} + \left(118 a + 37\right)\cdot 131^{4} +O\left(131^{ 5 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 6 }$

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character value
$1$$1$$()$$5$
$15$$2$$(1,2)(3,4)(5,6)$$-1$
$15$$2$$(1,2)$$3$
$45$$2$$(1,2)(3,4)$$1$
$40$$3$$(1,2,3)(4,5,6)$$-1$
$40$$3$$(1,2,3)$$2$
$90$$4$$(1,2,3,4)(5,6)$$-1$
$90$$4$$(1,2,3,4)$$1$
$144$$5$$(1,2,3,4,5)$$0$
$120$$6$$(1,2,3,4,5,6)$$-1$
$120$$6$$(1,2,3)(4,5)$$0$
The blue line marks the conjugacy class containing complex conjugation.