Properties

Label 5.46757.6t16.1c1
Dimension 5
Group $S_6$
Conductor $ 46757 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$5$
Group:$S_6$
Conductor:$46757 $
Artin number field: Splitting field of $f= x^{6} - x^{5} + 2 x^{3} - 2 x^{2} + 2 x - 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_6$
Parity: Even
Determinant: 1.46757.2t1.1c1

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:
$r_{ 1 }$ $=$ $ 13 a + 7 + \left(61 a + 18\right)\cdot 97 + \left(73 a + 48\right)\cdot 97^{2} + \left(45 a + 91\right)\cdot 97^{3} + \left(35 a + 15\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 80 + 52\cdot 97 + 10\cdot 97^{2} + 73\cdot 97^{3} + 93\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 79 a + 82 + \left(29 a + 45\right)\cdot 97 + \left(50 a + 48\right)\cdot 97^{2} + \left(2 a + 11\right)\cdot 97^{3} + \left(42 a + 86\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 84 a + 20 + \left(35 a + 66\right)\cdot 97 + \left(23 a + 60\right)\cdot 97^{2} + \left(51 a + 63\right)\cdot 97^{3} + \left(61 a + 5\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 18 a + 64 + \left(67 a + 93\right)\cdot 97 + \left(46 a + 68\right)\cdot 97^{2} + \left(94 a + 60\right)\cdot 97^{3} + \left(54 a + 28\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 39 + 14\cdot 97 + 54\cdot 97^{2} + 87\cdot 97^{3} + 60\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$$1$$()$$5$
$15$$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.