Properties

Label 5.3e10_13e4.6t15.1c1
Dimension 5
Group $A_6$
Conductor $ 3^{10} \cdot 13^{4}$
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$5$
Group:$A_6$
Conductor:$1686498489= 3^{10} \cdot 13^{4} $
Artin number field: Splitting field of $f= x^{6} - 3 x^{5} + 3 x^{4} - 3 x^{2} + 3 x + 2 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $A_6$
Parity: Even
Determinant: 1.1.1t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 151 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 151 }$: $ x^{2} + 149 x + 6 $
Roots:
$r_{ 1 }$ $=$ $ 148 a + 108 + \left(5 a + 33\right)\cdot 151 + \left(91 a + 103\right)\cdot 151^{2} + \left(100 a + 69\right)\cdot 151^{3} + \left(109 a + 82\right)\cdot 151^{4} +O\left(151^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 90 a + 21 + \left(91 a + 142\right)\cdot 151 + \left(6 a + 19\right)\cdot 151^{2} + \left(39 a + 64\right)\cdot 151^{3} + \left(35 a + 125\right)\cdot 151^{4} +O\left(151^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 61 a + 50 + \left(59 a + 84\right)\cdot 151 + \left(144 a + 92\right)\cdot 151^{2} + \left(111 a + 135\right)\cdot 151^{3} + \left(115 a + 5\right)\cdot 151^{4} +O\left(151^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 3 a + 102 + \left(145 a + 48\right)\cdot 151 + \left(59 a + 128\right)\cdot 151^{2} + \left(50 a + 28\right)\cdot 151^{3} + \left(41 a + 50\right)\cdot 151^{4} +O\left(151^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 101 + 14\cdot 151 + 73\cdot 151^{2} + 32\cdot 151^{3} + 81\cdot 151^{4} +O\left(151^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 74 + 129\cdot 151 + 35\cdot 151^{2} + 122\cdot 151^{3} + 107\cdot 151^{4} +O\left(151^{ 5 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 6 }$

Cycle notation
$(1,2,3)$
$(1,2)(3,4,5,6)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character value
$1$$1$$()$$5$
$45$$2$$(1,2)(3,4)$$1$
$40$$3$$(1,2,3)(4,5,6)$$2$
$40$$3$$(1,2,3)$$-1$
$90$$4$$(1,2,3,4)(5,6)$$-1$
$72$$5$$(1,2,3,4,5)$$0$
$72$$5$$(1,3,4,5,2)$$0$
The blue line marks the conjugacy class containing complex conjugation.