Properties

Label 5.2e12_3e10.6t15.3c1
Dimension 5
Group $A_6$
Conductor $ 2^{12} \cdot 3^{10}$
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$5$
Group:$A_6$
Conductor:$241864704= 2^{12} \cdot 3^{10} $
Artin number field: Splitting field of $f= x^{6} - 8 x^{3} + 9 x^{2} - 6 $ 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 $\Q_{ 113 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 113 }$: $ x^{2} + 101 x + 3 $
Roots:
$r_{ 1 }$ $=$ $ 107 a + 64 + \left(98 a + 25\right)\cdot 113 + \left(37 a + 54\right)\cdot 113^{2} + \left(55 a + 87\right)\cdot 113^{3} + \left(63 a + 38\right)\cdot 113^{4} +O\left(113^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 18 + 49\cdot 113 + 74\cdot 113^{2} + 76\cdot 113^{3} + 99\cdot 113^{4} +O\left(113^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 6 a + 105 + \left(14 a + 88\right)\cdot 113 + \left(75 a + 70\right)\cdot 113^{2} + \left(57 a + 35\right)\cdot 113^{3} + \left(49 a + 67\right)\cdot 113^{4} +O\left(113^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 10 a + 18 + \left(49 a + 26\right)\cdot 113 + \left(54 a + 32\right)\cdot 113^{2} + \left(44 a + 103\right)\cdot 113^{3} + \left(62 a + 75\right)\cdot 113^{4} +O\left(113^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 109 + 108\cdot 113 + 35\cdot 113^{2} + 18\cdot 113^{3} + 68\cdot 113^{4} +O\left(113^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 103 a + 25 + \left(63 a + 40\right)\cdot 113 + \left(58 a + 71\right)\cdot 113^{2} + \left(68 a + 17\right)\cdot 113^{3} + \left(50 a + 102\right)\cdot 113^{4} +O\left(113^{ 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.