Properties

Label 10.2e18_3e12_5e8.30t176.3c1
Dimension 10
Group $S_6$
Conductor $ 2^{18} \cdot 3^{12} \cdot 5^{8}$
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$10$
Group:$S_6$
Conductor:$54419558400000000= 2^{18} \cdot 3^{12} \cdot 5^{8} $
Artin number field: Splitting field of $f= x^{6} + 2 x^{4} - 2 x^{3} - 3 x^{2} - 6 x - 3 $ 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 $\Q_{ 137 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 137 }$: $ x^{2} + 131 x + 3 $
Roots:
$r_{ 1 }$ $=$ $ 41 a + 70 + \left(43 a + 31\right)\cdot 137 + \left(73 a + 12\right)\cdot 137^{2} + \left(136 a + 9\right)\cdot 137^{3} + \left(94 a + 98\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 116 a + 57 + \left(90 a + 11\right)\cdot 137 + \left(14 a + 57\right)\cdot 137^{2} + \left(99 a + 99\right)\cdot 137^{3} + \left(66 a + 19\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 102 a + 55 + \left(101 a + 132\right)\cdot 137 + \left(21 a + 61\right)\cdot 137^{2} + \left(60 a + 86\right)\cdot 137^{3} + \left(44 a + 96\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 35 a + 119 + \left(35 a + 92\right)\cdot 137 + \left(115 a + 90\right)\cdot 137^{2} + \left(76 a + 14\right)\cdot 137^{3} + \left(92 a + 29\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 96 a + 42 + \left(93 a + 113\right)\cdot 137 + \left(63 a + 134\right)\cdot 137^{2} + 69\cdot 137^{3} + \left(42 a + 120\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 21 a + 68 + \left(46 a + 29\right)\cdot 137 + \left(122 a + 54\right)\cdot 137^{2} + \left(37 a + 131\right)\cdot 137^{3} + \left(70 a + 46\right)\cdot 137^{4} +O\left(137^{ 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.