Properties

Label 5.2e4_7e4_31e2.6t15.1c1
Dimension 5
Group $A_6$
Conductor $ 2^{4} \cdot 7^{4} \cdot 31^{2}$
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$5$
Group:$A_6$
Conductor:$36917776= 2^{4} \cdot 7^{4} \cdot 31^{2} $
Artin number field: Splitting field of $f= x^{6} - 2 x^{5} + x^{4} - 2 x^{3} - x^{2} + 6 x - 1 $ 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 $\Q_{ 137 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 137 }$: $ x^{2} + 131 x + 3 $
Roots:
$r_{ 1 }$ $=$ $ 18 + 133\cdot 137 + 23\cdot 137^{2} + 120\cdot 137^{3} + 75\cdot 137^{4} +O\left(137^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 88 a + 19 + \left(119 a + 49\right)\cdot 137 + \left(95 a + 82\right)\cdot 137^{2} + \left(93 a + 123\right)\cdot 137^{3} + \left(82 a + 36\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 49 a + 136 + \left(17 a + 130\right)\cdot 137 + \left(41 a + 126\right)\cdot 137^{2} + \left(43 a + 41\right)\cdot 137^{3} + \left(54 a + 28\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 126 a + 6 + \left(61 a + 124\right)\cdot 137 + \left(2 a + 59\right)\cdot 137^{2} + \left(69 a + 105\right)\cdot 137^{3} + \left(105 a + 7\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 11 a + 77 + \left(75 a + 95\right)\cdot 137 + \left(134 a + 12\right)\cdot 137^{2} + \left(67 a + 106\right)\cdot 137^{3} + \left(31 a + 23\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 20 + 15\cdot 137 + 105\cdot 137^{2} + 50\cdot 137^{3} + 101\cdot 137^{4} +O\left(137^{ 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.