Properties

Label 5.101_431.6t16.1c1
Dimension 5
Group $S_6$
Conductor $ 101 \cdot 431 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$5$
Group:$S_6$
Conductor:$43531= 101 \cdot 431 $
Artin number field: Splitting field of $f= x^{6} - x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_6$
Parity: Odd
Determinant: 1.101_431.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 157 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 157 }$: $ x^{2} + 152 x + 5 $
Roots:
$r_{ 1 }$ $=$ $ 119 + 95\cdot 157 + 44\cdot 157^{2} + 72\cdot 157^{3} + 99\cdot 157^{4} +O\left(157^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 148 a + 40 + \left(95 a + 38\right)\cdot 157 + \left(94 a + 121\right)\cdot 157^{2} + \left(106 a + 72\right)\cdot 157^{3} + \left(131 a + 97\right)\cdot 157^{4} +O\left(157^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 59 + 63\cdot 157 + 5\cdot 157^{2} + 131\cdot 157^{3} + 46\cdot 157^{4} +O\left(157^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 25 a + 145 + \left(88 a + 57\right)\cdot 157 + \left(107 a + 68\right)\cdot 157^{2} + \left(82 a + 81\right)\cdot 157^{3} + \left(114 a + 93\right)\cdot 157^{4} +O\left(157^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 9 a + 152 + \left(61 a + 55\right)\cdot 157 + \left(62 a + 27\right)\cdot 157^{2} + \left(50 a + 40\right)\cdot 157^{3} + \left(25 a + 21\right)\cdot 157^{4} +O\left(157^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 132 a + 113 + \left(68 a + 2\right)\cdot 157 + \left(49 a + 47\right)\cdot 157^{2} + \left(74 a + 73\right)\cdot 157^{3} + \left(42 a + 112\right)\cdot 157^{4} +O\left(157^{ 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.