Properties

Label 5.2e3_3851.6t16.1c1
Dimension 5
Group $S_6$
Conductor $ 2^{3} \cdot 3851 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$5$
Group:$S_6$
Conductor:$30808= 2^{3} \cdot 3851 $
Artin number field: Splitting field of $f= x^{6} - x^{4} - x^{3} + x^{2} + x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_6$
Parity: Odd
Determinant: 1.2e3_3851.2t1.1c1

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:
$r_{ 1 }$ $=$ $ 59 a + 99 + \left(77 a + 33\right)\cdot 109 + 108 a\cdot 109^{2} + \left(71 a + 26\right)\cdot 109^{3} + \left(84 a + 78\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 20 a + 102 + \left(29 a + 107\right)\cdot 109 + \left(67 a + 17\right)\cdot 109^{2} + \left(18 a + 7\right)\cdot 109^{3} + \left(9 a + 77\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 89 a + 13 + \left(79 a + 8\right)\cdot 109 + \left(41 a + 56\right)\cdot 109^{2} + \left(90 a + 67\right)\cdot 109^{3} + \left(99 a + 67\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 50 a + 49 + \left(31 a + 52\right)\cdot 109 + 31\cdot 109^{2} + \left(37 a + 98\right)\cdot 109^{3} + \left(24 a + 90\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 3 a + 85 + \left(62 a + 32\right)\cdot 109 + \left(80 a + 101\right)\cdot 109^{2} + \left(3 a + 47\right)\cdot 109^{3} + \left(83 a + 21\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 106 a + 88 + \left(46 a + 91\right)\cdot 109 + \left(28 a + 10\right)\cdot 109^{2} + \left(105 a + 80\right)\cdot 109^{3} + \left(25 a + 100\right)\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 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.