Properties

Label 5.29077.6t16.1c1
Dimension 5
Group $S_6$
Conductor $ 29077 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$5$
Group:$S_6$
Conductor:$29077 $
Artin number field: Splitting field of $f= x^{6} - x^{5} + x^{4} - x^{2} + 2 x - 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_6$
Parity: Even
Determinant: 1.29077.2t1.1c1

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:
$r_{ 1 }$ $=$ $ 8 a + 129 + \left(55 a + 128\right)\cdot 131 + \left(78 a + 32\right)\cdot 131^{2} + \left(19 a + 108\right)\cdot 131^{3} + \left(57 a + 50\right)\cdot 131^{4} +O\left(131^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 123 a + 30 + \left(75 a + 79\right)\cdot 131 + \left(52 a + 29\right)\cdot 131^{2} + \left(111 a + 108\right)\cdot 131^{3} + \left(73 a + 128\right)\cdot 131^{4} +O\left(131^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 111 a + 49 + \left(54 a + 105\right)\cdot 131 + \left(61 a + 111\right)\cdot 131^{2} + \left(18 a + 53\right)\cdot 131^{3} + \left(13 a + 4\right)\cdot 131^{4} +O\left(131^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 55 a + 64 + \left(70 a + 81\right)\cdot 131 + \left(109 a + 101\right)\cdot 131^{2} + \left(126 a + 25\right)\cdot 131^{3} + \left(24 a + 33\right)\cdot 131^{4} +O\left(131^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 76 a + 22 + \left(60 a + 46\right)\cdot 131 + \left(21 a + 76\right)\cdot 131^{2} + \left(4 a + 30\right)\cdot 131^{3} + \left(106 a + 6\right)\cdot 131^{4} +O\left(131^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 20 a + 100 + \left(76 a + 82\right)\cdot 131 + \left(69 a + 40\right)\cdot 131^{2} + \left(112 a + 66\right)\cdot 131^{3} + \left(117 a + 38\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.