Properties

Label 5.47933.6t16.1
Dimension 5
Group $S_6$
Conductor $ 47933 $
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$5$
Group:$S_6$
Conductor:$47933 $
Artin number field: Splitting field of $f= x^{6} - 2 x^{5} + 2 x^{4} - x^{3} - 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_6$
Parity: Even

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 109 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 109 }$: $ x^{2} + 108 x + 6 $
Roots:
$r_{ 1 }$ $=$ $ 82 a + 7 + \left(41 a + 33\right)\cdot 109 + \left(70 a + 36\right)\cdot 109^{2} + \left(13 a + 104\right)\cdot 109^{3} + \left(76 a + 97\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 80 a + 41 + \left(101 a + 78\right)\cdot 109 + \left(21 a + 102\right)\cdot 109^{2} + \left(89 a + 5\right)\cdot 109^{3} + \left(97 a + 35\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 29 a + 12 + \left(7 a + 100\right)\cdot 109 + \left(87 a + 22\right)\cdot 109^{2} + \left(19 a + 73\right)\cdot 109^{3} + \left(11 a + 43\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 27 a + 89 + \left(67 a + 101\right)\cdot 109 + \left(38 a + 64\right)\cdot 109^{2} + \left(95 a + 47\right)\cdot 109^{3} + \left(32 a + 51\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 21 + 102\cdot 109 + 63\cdot 109^{2} + 60\cdot 109^{3} + 60\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 50 + 20\cdot 109 + 36\cdot 109^{2} + 35\cdot 109^{3} + 38\cdot 109^{4} +O\left(109^{ 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 values
$c1$
$1$ $1$ $()$ $5$
$15$ $2$ $(1,2)(3,4)(5,6)$ $-1$
$15$ $2$ $(1,2)$ $3$
$45$ $2$ $(1,2)(3,4)$ $1$
$40$ $3$ $(1,2,3)(4,5,6)$ $-1$
$40$ $3$ $(1,2,3)$ $2$
$90$ $4$ $(1,2,3,4)(5,6)$ $-1$
$90$ $4$ $(1,2,3,4)$ $1$
$144$ $5$ $(1,2,3,4,5)$ $0$
$120$ $6$ $(1,2,3,4,5,6)$ $-1$
$120$ $6$ $(1,2,3)(4,5)$ $0$
The blue line marks the conjugacy class containing complex conjugation.