Properties

Label 10.3e14_193e4.30t176.1c1
Dimension 10
Group $S_6$
Conductor $ 3^{14} \cdot 193^{4}$
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$10$
Group:$S_6$
Conductor:$6636312096654969= 3^{14} \cdot 193^{4} $
Artin number field: Splitting field of $f= x^{6} - 2 x^{3} - 3 x^{2} - 3 x - 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: 30T176
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:
$r_{ 1 }$ $=$ $ 33 a + 67 + \left(92 a + 35\right)\cdot 137 + \left(101 a + 9\right)\cdot 137^{2} + \left(16 a + 42\right)\cdot 137^{3} + \left(84 a + 132\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 10 a + 9 + \left(37 a + 97\right)\cdot 137 + \left(34 a + 50\right)\cdot 137^{2} + \left(11 a + 30\right)\cdot 137^{3} + \left(128 a + 128\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 127 a + 69 + \left(99 a + 35\right)\cdot 137 + \left(102 a + 82\right)\cdot 137^{2} + \left(125 a + 63\right)\cdot 137^{3} + \left(8 a + 63\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 43 + 8\cdot 137 + 92\cdot 137^{2} + 70\cdot 137^{3} + 74\cdot 137^{4} +O\left(137^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 104 a + 128 + \left(44 a + 7\right)\cdot 137 + \left(35 a + 116\right)\cdot 137^{2} + \left(120 a + 40\right)\cdot 137^{3} + \left(52 a + 72\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 95 + 89\cdot 137 + 60\cdot 137^{2} + 26\cdot 137^{3} + 77\cdot 137^{4} +O\left(137^{ 5 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 6 }$

Cycle notation
$(1,2)$
$(1,2,3,4,5,6)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character value
$1$$1$$()$$10$
$15$$2$$(1,2)(3,4)(5,6)$$-2$
$15$$2$$(1,2)$$2$
$45$$2$$(1,2)(3,4)$$-2$
$40$$3$$(1,2,3)(4,5,6)$$1$
$40$$3$$(1,2,3)$$1$
$90$$4$$(1,2,3,4)(5,6)$$0$
$90$$4$$(1,2,3,4)$$0$
$144$$5$$(1,2,3,4,5)$$0$
$120$$6$$(1,2,3,4,5,6)$$1$
$120$$6$$(1,2,3)(4,5)$$-1$
The blue line marks the conjugacy class containing complex conjugation.