Properties

Label 10.139...000.30t164.a.a
Dimension $10$
Group $S_6$
Conductor $1.395\times 10^{16}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $10$
Group: $S_6$
Conductor: \(13947137604000000\)\(\medspace = 2^{8} \cdot 3^{20} \cdot 5^{6} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 6.2.4723920.2
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 6.2.4723920.2

Defining polynomial

$f(x)$$=$ \( x^{6} + 3x^{4} - 4x^{3} + 9x^{2} - 6x + 1 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 149 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 149 }$: \( x^{2} + 145x + 2 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 102 a + 4 + \left(126 a + 81\right)\cdot 149 + \left(113 a + 123\right)\cdot 149^{2} + \left(88 a + 112\right)\cdot 149^{3} + \left(125 a + 8\right)\cdot 149^{4} +O(149^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 47 a + 114 + \left(22 a + 38\right)\cdot 149 + \left(35 a + 5\right)\cdot 149^{2} + \left(60 a + 56\right)\cdot 149^{3} + \left(23 a + 124\right)\cdot 149^{4} +O(149^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 87 a + 64 + \left(103 a + 96\right)\cdot 149 + \left(57 a + 98\right)\cdot 149^{2} + \left(62 a + 136\right)\cdot 149^{3} + \left(125 a + 85\right)\cdot 149^{4} +O(149^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 62 a + 114 + \left(45 a + 125\right)\cdot 149 + \left(91 a + 76\right)\cdot 149^{2} + \left(86 a + 30\right)\cdot 149^{3} + \left(23 a + 78\right)\cdot 149^{4} +O(149^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 104 + 57\cdot 149 + 6\cdot 149^{2} + 90\cdot 149^{3} + 85\cdot 149^{4} +O(149^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 47 + 47\cdot 149 + 136\cdot 149^{2} + 20\cdot 149^{3} + 64\cdot 149^{4} +O(149^{5})\) 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.