Properties

Label 16.544...000.36t1252.a.a
Dimension $16$
Group $S_6$
Conductor $5.448\times 10^{25}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $16$
Group: $S_6$
Conductor: \(544\!\cdots\!000\)\(\medspace = 2^{12} \cdot 3^{20} \cdot 5^{18}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 6.2.9112500.1
Galois orbit size: $1$
Smallest permutation container: 36T1252
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $S_6$
Projective stem field: Galois closure of 6.2.9112500.1

Defining polynomial

$f(x)$$=$ \( x^{6} - 3x^{5} + 5x^{3} - 5 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 157 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 157 }$: \( x^{2} + 152x + 5 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 119 + 129\cdot 157 + 27\cdot 157^{2} + 55\cdot 157^{3} + 65\cdot 157^{4} +O(157^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 137 + 77\cdot 157 + 64\cdot 157^{2} + 115\cdot 157^{3} + 25\cdot 157^{4} +O(157^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 139 + 29\cdot 157 + 32\cdot 157^{2} + 97\cdot 157^{3} + 6\cdot 157^{4} +O(157^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 156 a + 20 + \left(84 a + 66\right)\cdot 157 + \left(15 a + 146\right)\cdot 157^{2} + \left(56 a + 142\right)\cdot 157^{3} + \left(55 a + 78\right)\cdot 157^{4} +O(157^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( a + 15 + \left(72 a + 21\right)\cdot 157 + \left(141 a + 139\right)\cdot 157^{2} + \left(100 a + 93\right)\cdot 157^{3} + \left(101 a + 142\right)\cdot 157^{4} +O(157^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 44 + 146\cdot 157 + 60\cdot 157^{2} + 123\cdot 157^{3} + 151\cdot 157^{4} +O(157^{5})\) 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$$1$$()$$16$
$15$$2$$(1,2)(3,4)(5,6)$$0$
$15$$2$$(1,2)$$0$
$45$$2$$(1,2)(3,4)$$0$
$40$$3$$(1,2,3)(4,5,6)$$-2$
$40$$3$$(1,2,3)$$-2$
$90$$4$$(1,2,3,4)(5,6)$$0$
$90$$4$$(1,2,3,4)$$0$
$144$$5$$(1,2,3,4,5)$$1$
$120$$6$$(1,2,3,4,5,6)$$0$
$120$$6$$(1,2,3)(4,5)$$0$

The blue line marks the conjugacy class containing complex conjugation.