Properties

Label 5.2e14_3e10.6t15.1c1
Dimension 5
Group $A_6$
Conductor $ 2^{14} \cdot 3^{10}$
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$5$
Group:$A_6$
Conductor:$967458816= 2^{14} \cdot 3^{10} $
Artin number field: Splitting field of $f= x^{6} - 12 x^{3} + 21 x^{2} + 12 x - 34 $ 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 $\Q_{ 47 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: $ x^{2} + 45 x + 5 $
Roots:
$r_{ 1 }$ $=$ $ 4 a + 30 + \left(18 a + 31\right)\cdot 47 + \left(2 a + 30\right)\cdot 47^{2} + \left(14 a + 4\right)\cdot 47^{3} + \left(6 a + 22\right)\cdot 47^{4} + \left(34 a + 16\right)\cdot 47^{5} +O\left(47^{ 6 }\right)$
$r_{ 2 }$ $=$ $ 5 a + 35 + \left(16 a + 43\right)\cdot 47 + \left(43 a + 42\right)\cdot 47^{2} + \left(28 a + 11\right)\cdot 47^{3} + \left(18 a + 16\right)\cdot 47^{4} + \left(a + 21\right)\cdot 47^{5} +O\left(47^{ 6 }\right)$
$r_{ 3 }$ $=$ $ 16 + 12\cdot 47 + 22\cdot 47^{2} + 20\cdot 47^{3} + 13\cdot 47^{4} + 47^{5} +O\left(47^{ 6 }\right)$
$r_{ 4 }$ $=$ $ 24 + 12\cdot 47 + 8\cdot 47^{2} + 44\cdot 47^{4} + 17\cdot 47^{5} +O\left(47^{ 6 }\right)$
$r_{ 5 }$ $=$ $ 42 a + 45 + \left(30 a + 23\right)\cdot 47 + \left(3 a + 19\right)\cdot 47^{2} + \left(18 a + 26\right)\cdot 47^{3} + \left(28 a + 24\right)\cdot 47^{4} + \left(45 a + 5\right)\cdot 47^{5} +O\left(47^{ 6 }\right)$
$r_{ 6 }$ $=$ $ 43 a + 38 + \left(28 a + 16\right)\cdot 47 + \left(44 a + 17\right)\cdot 47^{2} + \left(32 a + 30\right)\cdot 47^{3} + \left(40 a + 20\right)\cdot 47^{4} + \left(12 a + 31\right)\cdot 47^{5} +O\left(47^{ 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.