Properties

Label 5.1108809.6t15.a.a
Dimension $5$
Group $A_6$
Conductor $1108809$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $5$
Group: $A_6$
Conductor: \(1108809\)\(\medspace = 3^{8} \cdot 13^{2} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 6.2.1108809.1
Galois orbit size: $1$
Smallest permutation container: $A_6$
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $A_6$
Projective stem field: Galois closure of 6.2.1108809.1

Defining polynomial

$f(x)$$=$ \( x^{6} - 3x^{5} + 3x^{4} - 3x^{2} + 3x + 2 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 151 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 151 }$: \( x^{2} + 149x + 6 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 148 a + 108 + \left(5 a + 33\right)\cdot 151 + \left(91 a + 103\right)\cdot 151^{2} + \left(100 a + 69\right)\cdot 151^{3} + \left(109 a + 82\right)\cdot 151^{4} +O(151^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 90 a + 21 + \left(91 a + 142\right)\cdot 151 + \left(6 a + 19\right)\cdot 151^{2} + \left(39 a + 64\right)\cdot 151^{3} + \left(35 a + 125\right)\cdot 151^{4} +O(151^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 61 a + 50 + \left(59 a + 84\right)\cdot 151 + \left(144 a + 92\right)\cdot 151^{2} + \left(111 a + 135\right)\cdot 151^{3} + \left(115 a + 5\right)\cdot 151^{4} +O(151^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 3 a + 102 + \left(145 a + 48\right)\cdot 151 + \left(59 a + 128\right)\cdot 151^{2} + \left(50 a + 28\right)\cdot 151^{3} + \left(41 a + 50\right)\cdot 151^{4} +O(151^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 101 + 14\cdot 151 + 73\cdot 151^{2} + 32\cdot 151^{3} + 81\cdot 151^{4} +O(151^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 74 + 129\cdot 151 + 35\cdot 151^{2} + 122\cdot 151^{3} + 107\cdot 151^{4} +O(151^{5})\) Copy content Toggle raw display

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)$$-1$
$40$$3$$(1,2,3)$$2$
$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.