Properties

Label 5.36107.6t16.1c1
Dimension 5
Group $S_6$
Conductor $ 36107 $
Root number 1
Frobenius-Schur indicator 1

Related objects

Learn more about

Basic invariants

Dimension:$5$
Group:$S_6$
Conductor:$36107 $
Artin number field: Splitting field of $f= x^{6} - 2 x^{5} + x^{4} - x^{2} + x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_6$
Parity: Odd
Determinant: 1.36107.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 131 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 131 }$: $ x^{2} + 127 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 57 a + 61 + \left(110 a + 39\right)\cdot 131 + \left(68 a + 97\right)\cdot 131^{2} + \left(2 a + 75\right)\cdot 131^{3} + \left(80 a + 56\right)\cdot 131^{4} +O\left(131^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 74 a + 27 + \left(20 a + 31\right)\cdot 131 + 62 a\cdot 131^{2} + \left(128 a + 17\right)\cdot 131^{3} + \left(50 a + 112\right)\cdot 131^{4} +O\left(131^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 62 a + 109 + \left(56 a + 108\right)\cdot 131 + \left(97 a + 94\right)\cdot 131^{2} + \left(29 a + 76\right)\cdot 131^{3} + \left(53 a + 99\right)\cdot 131^{4} +O\left(131^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 69 a + 95 + \left(74 a + 10\right)\cdot 131 + \left(33 a + 35\right)\cdot 131^{2} + \left(101 a + 98\right)\cdot 131^{3} + \left(77 a + 20\right)\cdot 131^{4} +O\left(131^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 24 + 92\cdot 131 + 97\cdot 131^{2} + 96\cdot 131^{3} + 91\cdot 131^{4} +O\left(131^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 79 + 110\cdot 131 + 67\cdot 131^{2} + 28\cdot 131^{3} + 12\cdot 131^{4} +O\left(131^{ 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.