Properties

Label 10.2e24_3e8_5e6.30t176.3c1
Dimension 10
Group $S_6$
Conductor $ 2^{24} \cdot 3^{8} \cdot 5^{6}$
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$10$
Group:$S_6$
Conductor:$1719926784000000= 2^{24} \cdot 3^{8} \cdot 5^{6} $
Artin number field: Splitting field of $f= x^{6} - 2 x^{5} - x^{4} + 6 x^{3} - 2 x^{2} - 4 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 $\Q_{ 137 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 137 }$: $ x^{2} + 131 x + 3 $
Roots:
$r_{ 1 }$ $=$ $ 118 a + 5 + \left(93 a + 77\right)\cdot 137 + \left(97 a + 36\right)\cdot 137^{2} + \left(50 a + 102\right)\cdot 137^{3} + \left(121 a + 19\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 19 a + 28 + \left(43 a + 111\right)\cdot 137 + \left(39 a + 117\right)\cdot 137^{2} + \left(86 a + 34\right)\cdot 137^{3} + \left(15 a + 12\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 51 + 99\cdot 137 + 51\cdot 137^{2} + 46\cdot 137^{3} + 97\cdot 137^{4} +O\left(137^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 29 a + 88 + \left(21 a + 131\right)\cdot 137 + \left(51 a + 24\right)\cdot 137^{2} + \left(96 a + 68\right)\cdot 137^{3} + \left(91 a + 92\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 116 + 35\cdot 137 + 6\cdot 137^{2} + 112\cdot 137^{3} + 53\cdot 137^{4} +O\left(137^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 108 a + 125 + \left(115 a + 92\right)\cdot 137 + \left(85 a + 36\right)\cdot 137^{2} + \left(40 a + 47\right)\cdot 137^{3} + \left(45 a + 135\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.