Properties

Label 5.3e10_5e8.6t15.1c1
Dimension 5
Group $A_6$
Conductor $ 3^{10} \cdot 5^{8}$
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$5$
Group:$A_6$
Conductor:$23066015625= 3^{10} \cdot 5^{8} $
Artin number field: Splitting field of $f= x^{6} - 5 x^{3} + 45 x^{2} - 99 x - 15 $ 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 $\Q_{ 127 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 127 }$: $ x^{2} + 126 x + 3 $
Roots:
$r_{ 1 }$ $=$ $ 95 a + 57 + \left(10 a + 13\right)\cdot 127 + \left(10 a + 29\right)\cdot 127^{2} + \left(23 a + 48\right)\cdot 127^{3} + \left(17 a + 2\right)\cdot 127^{4} + \left(2 a + 31\right)\cdot 127^{5} +O\left(127^{ 6 }\right)$
$r_{ 2 }$ $=$ $ 15 + 55\cdot 127 + 61\cdot 127^{2} + 79\cdot 127^{3} + 35\cdot 127^{4} + 7\cdot 127^{5} +O\left(127^{ 6 }\right)$
$r_{ 3 }$ $=$ $ 32 a + 25 + \left(116 a + 56\right)\cdot 127 + \left(116 a + 28\right)\cdot 127^{2} + \left(103 a + 61\right)\cdot 127^{3} + \left(109 a + 123\right)\cdot 127^{4} + \left(124 a + 15\right)\cdot 127^{5} +O\left(127^{ 6 }\right)$
$r_{ 4 }$ $=$ $ 72 + 79\cdot 127 + 121\cdot 127^{2} + 51\cdot 127^{3} + 67\cdot 127^{4} + 51\cdot 127^{5} +O\left(127^{ 6 }\right)$
$r_{ 5 }$ $=$ $ 94 a + 59 + \left(40 a + 51\right)\cdot 127 + \left(71 a + 118\right)\cdot 127^{2} + \left(33 a + 88\right)\cdot 127^{3} + \left(63 a + 124\right)\cdot 127^{4} + \left(3 a + 103\right)\cdot 127^{5} +O\left(127^{ 6 }\right)$
$r_{ 6 }$ $=$ $ 33 a + 26 + \left(86 a + 125\right)\cdot 127 + \left(55 a + 21\right)\cdot 127^{2} + \left(93 a + 51\right)\cdot 127^{3} + \left(63 a + 27\right)\cdot 127^{4} + \left(123 a + 44\right)\cdot 127^{5} +O\left(127^{ 6 }\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.