Properties

Label 5.37463.6t16.1c1
Dimension 5
Group $S_6$
Conductor $ 37463 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$5$
Group:$S_6$
Conductor:$37463 $
Artin number field: Splitting field of $f= x^{6} - x^{4} - x^{3} + 2 x^{2} + 2 x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_6$
Parity: Odd
Determinant: 1.37463.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 167 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 167 }$: $ x^{2} + 166 x + 5 $
Roots:
$r_{ 1 }$ $=$ $ 87 + 12\cdot 167 + 50\cdot 167^{2} + 43\cdot 167^{3} + 126\cdot 167^{4} +O\left(167^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 164 a + 153 + \left(76 a + 64\right)\cdot 167 + \left(157 a + 3\right)\cdot 167^{2} + \left(102 a + 165\right)\cdot 167^{3} + \left(12 a + 120\right)\cdot 167^{4} +O\left(167^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 57 a + 23 + \left(21 a + 63\right)\cdot 167 + \left(41 a + 142\right)\cdot 167^{2} + \left(47 a + 1\right)\cdot 167^{3} + \left(16 a + 42\right)\cdot 167^{4} +O\left(167^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 8 + 21\cdot 167 + 59\cdot 167^{2} + 5\cdot 167^{3} + 3\cdot 167^{4} +O\left(167^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 3 a + 150 + \left(90 a + 144\right)\cdot 167 + \left(9 a + 83\right)\cdot 167^{2} + \left(64 a + 110\right)\cdot 167^{3} + \left(154 a + 30\right)\cdot 167^{4} +O\left(167^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 110 a + 80 + \left(145 a + 27\right)\cdot 167 + \left(125 a + 162\right)\cdot 167^{2} + \left(119 a + 7\right)\cdot 167^{3} + \left(150 a + 11\right)\cdot 167^{4} +O\left(167^{ 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.