Properties

Label 16.129...000.36t1252.a.a
Dimension $16$
Group $S_6$
Conductor $1.297\times 10^{26}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $16$
Group: $S_6$
Conductor: \(129\!\cdots\!000\)\(\medspace = 2^{12} \cdot 3^{12} \cdot 5^{24}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 6.2.25312500.1
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 6.2.25312500.1

Defining polynomial

$f(x)$$=$ \( x^{6} - 3x^{5} + 5x^{3} + 12x + 4 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: \( x^{2} + 16x + 3 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 11 a + 10 + \left(3 a + 8\right)\cdot 17 + \left(10 a + 5\right)\cdot 17^{2} + 6\cdot 17^{3} + \left(8 a + 12\right)\cdot 17^{4} +O(17^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 15 a + 16 + \left(9 a + 1\right)\cdot 17 + \left(13 a + 10\right)\cdot 17^{2} + \left(6 a + 12\right)\cdot 17^{3} + \left(3 a + 16\right)\cdot 17^{4} +O(17^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 8 a + 1 + \left(7 a + 13\right)\cdot 17 + \left(a + 7\right)\cdot 17^{2} + \left(8 a + 11\right)\cdot 17^{3} + \left(a + 14\right)\cdot 17^{4} +O(17^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 6 a + 4 + \left(13 a + 1\right)\cdot 17 + \left(6 a + 12\right)\cdot 17^{2} + \left(16 a + 13\right)\cdot 17^{3} + \left(8 a + 2\right)\cdot 17^{4} +O(17^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 2 a + 14 + \left(7 a + 13\right)\cdot 17 + \left(3 a + 13\right)\cdot 17^{2} + \left(10 a + 5\right)\cdot 17^{3} + \left(13 a + 13\right)\cdot 17^{4} +O(17^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 9 a + 9 + \left(9 a + 12\right)\cdot 17 + \left(15 a + 1\right)\cdot 17^{2} + \left(8 a + 1\right)\cdot 17^{3} + \left(15 a + 8\right)\cdot 17^{4} +O(17^{5})\) 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.