Properties

Label 5.22291.6t16.1c1
Dimension 5
Group $S_6$
Conductor $ 22291 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$5$
Group:$S_6$
Conductor:$22291 $
Artin number field: Splitting field of $f= x^{6} + x^{4} - x^{3} - x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_6$
Parity: Odd
Determinant: 1.22291.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 113 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 113 }$: $ x^{2} + 101 x + 3 $
Roots:
$r_{ 1 }$ $=$ $ 60 a + 91 + \left(85 a + 84\right)\cdot 113 + \left(109 a + 82\right)\cdot 113^{2} + \left(109 a + 88\right)\cdot 113^{3} + \left(39 a + 65\right)\cdot 113^{4} +O\left(113^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 78 + 98\cdot 113 + 28\cdot 113^{2} + 66\cdot 113^{3} + 94\cdot 113^{4} +O\left(113^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 103 + 62\cdot 113 + 82\cdot 113^{2} + 56\cdot 113^{3} + 94\cdot 113^{4} +O\left(113^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 53 a + 20 + \left(27 a + 34\right)\cdot 113 + \left(3 a + 71\right)\cdot 113^{2} + \left(3 a + 55\right)\cdot 113^{3} + \left(73 a + 96\right)\cdot 113^{4} +O\left(113^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 9 a + 26 + \left(47 a + 90\right)\cdot 113 + \left(31 a + 97\right)\cdot 113^{2} + \left(21 a + 36\right)\cdot 113^{3} + \left(69 a + 41\right)\cdot 113^{4} +O\left(113^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 104 a + 21 + \left(65 a + 81\right)\cdot 113 + \left(81 a + 88\right)\cdot 113^{2} + \left(91 a + 34\right)\cdot 113^{3} + \left(43 a + 59\right)\cdot 113^{4} +O\left(113^{ 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.