Properties

Label 5.73_709.6t16.1
Dimension 5
Group $S_6$
Conductor $ 73 \cdot 709 $
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$5$
Group:$S_6$
Conductor:$51757= 73 \cdot 709 $
Artin number field: Splitting field of $f= x^{6} - 2 x^{3} - 2 x^{2} + x + 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 $\Q_{ 113 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 113 }$: $ x^{2} + 101 x + 3 $
Roots:
$r_{ 1 }$ $=$ $ 23 a + 80 + \left(43 a + 75\right)\cdot 113 + \left(104 a + 69\right)\cdot 113^{2} + \left(a + 98\right)\cdot 113^{3} + \left(29 a + 104\right)\cdot 113^{4} +O\left(113^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 74 a + 93 + \left(81 a + 50\right)\cdot 113 + \left(61 a + 28\right)\cdot 113^{2} + \left(101 a + 41\right)\cdot 113^{3} + \left(7 a + 22\right)\cdot 113^{4} +O\left(113^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 13 a + 71 + \left(5 a + 52\right)\cdot 113 + \left(54 a + 58\right)\cdot 113^{2} + \left(96 a + 13\right)\cdot 113^{3} + \left(96 a + 74\right)\cdot 113^{4} +O\left(113^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 39 a + 77 + \left(31 a + 52\right)\cdot 113 + \left(51 a + 9\right)\cdot 113^{2} + \left(11 a + 68\right)\cdot 113^{3} + \left(105 a + 15\right)\cdot 113^{4} +O\left(113^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 90 a + 17 + \left(69 a + 6\right)\cdot 113 + \left(8 a + 36\right)\cdot 113^{2} + \left(111 a + 17\right)\cdot 113^{3} + \left(83 a + 112\right)\cdot 113^{4} +O\left(113^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 100 a + 1 + \left(107 a + 101\right)\cdot 113 + \left(58 a + 23\right)\cdot 113^{2} + \left(16 a + 100\right)\cdot 113^{3} + \left(16 a + 9\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 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.