Properties

Label 16.809...641.36t1252.a.a
Dimension $16$
Group $S_6$
Conductor $8.093\times 10^{36}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $16$
Group: $S_6$
Conductor: \(809\!\cdots\!641\)\(\medspace = 7^{8} \cdot 5867^{8} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 6.2.41069.1
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 6.2.41069.1

Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: \( x^{2} + 33x + 2 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 2 + 20\cdot 37 + 15\cdot 37^{2} + 35\cdot 37^{3} + 19\cdot 37^{4} +O(37^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 13 a + 6 + \left(11 a + 2\right)\cdot 37 + \left(5 a + 23\right)\cdot 37^{2} + \left(3 a + 8\right)\cdot 37^{3} + \left(12 a + 22\right)\cdot 37^{4} +O(37^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 9 a + 26 + \left(28 a + 23\right)\cdot 37 + \left(11 a + 26\right)\cdot 37^{2} + \left(9 a + 30\right)\cdot 37^{3} + \left(22 a + 21\right)\cdot 37^{4} +O(37^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 31 + 13\cdot 37 + 4\cdot 37^{2} + 37^{3} + 26\cdot 37^{4} +O(37^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 28 a + 25 + \left(8 a + 16\right)\cdot 37 + \left(25 a + 8\right)\cdot 37^{2} + \left(27 a + 19\right)\cdot 37^{3} + \left(14 a + 27\right)\cdot 37^{4} +O(37^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 24 a + 21 + \left(25 a + 34\right)\cdot 37 + \left(31 a + 32\right)\cdot 37^{2} + \left(33 a + 15\right)\cdot 37^{3} + \left(24 a + 30\right)\cdot 37^{4} +O(37^{5})\) 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.