Properties

Label 10.2e26_3e16.30t176.4c1
Dimension 10
Group $S_6$
Conductor $ 2^{26} \cdot 3^{16}$
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$10$
Group:$S_6$
Conductor:$2888816545234944= 2^{26} \cdot 3^{16} $
Artin number field: Splitting field of $f= x^{6} - 2 x^{5} + x^{4} + x^{2} - 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 $\Q_{ 97 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 97 }$: $ x^{2} + 96 x + 5 $
Roots:
$r_{ 1 }$ $=$ $ 88 a + 69 + \left(96 a + 34\right)\cdot 97 + \left(71 a + 70\right)\cdot 97^{2} + \left(73 a + 14\right)\cdot 97^{3} + \left(64 a + 51\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 9 a + 60 + 43\cdot 97 + \left(25 a + 45\right)\cdot 97^{2} + \left(23 a + 16\right)\cdot 97^{3} + \left(32 a + 42\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 62 a + 10 + \left(77 a + 3\right)\cdot 97 + \left(31 a + 33\right)\cdot 97^{2} + \left(93 a + 42\right)\cdot 97^{3} + \left(41 a + 81\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 45 + 96\cdot 97 + 37\cdot 97^{2} + 86\cdot 97^{3} + 19\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 35 a + 72 + \left(19 a + 18\right)\cdot 97 + \left(65 a + 84\right)\cdot 97^{2} + \left(3 a + 6\right)\cdot 97^{3} + \left(55 a + 30\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 37 + 94\cdot 97 + 19\cdot 97^{2} + 27\cdot 97^{3} + 66\cdot 97^{4} +O\left(97^{ 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.