Properties

Label 10.663...969.30t164.a.a
Dimension $10$
Group $S_6$
Conductor $6.636\times 10^{15}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $10$
Group: $S_6$
Conductor: \(6636312096654969\)\(\medspace = 3^{14} \cdot 193^{4} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 6.2.140697.1
Galois orbit size: $1$
Smallest permutation container: 30T164
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $S_6$
Projective stem field: Galois closure of 6.2.140697.1

Defining polynomial

$f(x)$$=$ \( x^{6} - 2x^{3} - 3x^{2} - 3x - 1 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 137 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 137 }$: \( x^{2} + 131x + 3 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 33 a + 67 + \left(92 a + 35\right)\cdot 137 + \left(101 a + 9\right)\cdot 137^{2} + \left(16 a + 42\right)\cdot 137^{3} + \left(84 a + 132\right)\cdot 137^{4} +O(137^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 10 a + 9 + \left(37 a + 97\right)\cdot 137 + \left(34 a + 50\right)\cdot 137^{2} + \left(11 a + 30\right)\cdot 137^{3} + \left(128 a + 128\right)\cdot 137^{4} +O(137^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 127 a + 69 + \left(99 a + 35\right)\cdot 137 + \left(102 a + 82\right)\cdot 137^{2} + \left(125 a + 63\right)\cdot 137^{3} + \left(8 a + 63\right)\cdot 137^{4} +O(137^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 43 + 8\cdot 137 + 92\cdot 137^{2} + 70\cdot 137^{3} + 74\cdot 137^{4} +O(137^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 104 a + 128 + \left(44 a + 7\right)\cdot 137 + \left(35 a + 116\right)\cdot 137^{2} + \left(120 a + 40\right)\cdot 137^{3} + \left(52 a + 72\right)\cdot 137^{4} +O(137^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 95 + 89\cdot 137 + 60\cdot 137^{2} + 26\cdot 137^{3} + 77\cdot 137^{4} +O(137^{5})\) Copy content Toggle raw display

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.