Properties

Label 10.2e6_3e12_149e4.30t176.1c1
Dimension 10
Group $S_6$
Conductor $ 2^{6} \cdot 3^{12} \cdot 149^{4}$
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$10$
Group:$S_6$
Conductor:$16764094652917824= 2^{6} \cdot 3^{12} \cdot 149^{4} $
Artin number field: Splitting field of $f= x^{6} - x^{5} + 2 x^{4} + x^{3} - 2 x^{2} + x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: 30T176
Parity: Even
Determinant: 1.1.1t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 139 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 139 }$: $ x^{2} + 138 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 136 a + 125 + \left(110 a + 62\right)\cdot 139 + \left(135 a + 120\right)\cdot 139^{2} + \left(78 a + 121\right)\cdot 139^{3} + \left(43 a + 107\right)\cdot 139^{4} +O\left(139^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 3 a + 122 + \left(28 a + 37\right)\cdot 139 + \left(3 a + 6\right)\cdot 139^{2} + \left(60 a + 65\right)\cdot 139^{3} + \left(95 a + 72\right)\cdot 139^{4} +O\left(139^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 93 a + 72 + \left(12 a + 137\right)\cdot 139 + \left(7 a + 84\right)\cdot 139^{2} + \left(109 a + 52\right)\cdot 139^{3} + \left(55 a + 46\right)\cdot 139^{4} +O\left(139^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 22 a + 95 + \left(133 a + 74\right)\cdot 139 + \left(116 a + 1\right)\cdot 139^{2} + \left(2 a + 138\right)\cdot 139^{3} + \left(55 a + 2\right)\cdot 139^{4} +O\left(139^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 117 a + 117 + \left(5 a + 46\right)\cdot 139 + \left(22 a + 124\right)\cdot 139^{2} + \left(136 a + 23\right)\cdot 139^{3} + \left(83 a + 55\right)\cdot 139^{4} +O\left(139^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 46 a + 26 + \left(126 a + 57\right)\cdot 139 + \left(131 a + 79\right)\cdot 139^{2} + \left(29 a + 15\right)\cdot 139^{3} + \left(83 a + 132\right)\cdot 139^{4} +O\left(139^{ 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$$1$$()$$10$
$15$$2$$(1,2)(3,4)(5,6)$$-2$
$15$$2$$(1,2)$$2$
$45$$2$$(1,2)(3,4)$$-2$
$40$$3$$(1,2,3)(4,5,6)$$1$
$40$$3$$(1,2,3)$$1$
$90$$4$$(1,2,3,4)(5,6)$$0$
$90$$4$$(1,2,3,4)$$0$
$144$$5$$(1,2,3,4,5)$$0$
$120$$6$$(1,2,3,4,5,6)$$1$
$120$$6$$(1,2,3)(4,5)$$-1$
The blue line marks the conjugacy class containing complex conjugation.