Properties

Label 5.662908009.12t183.a.a
Dimension $5$
Group $S_6$
Conductor $662908009$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $5$
Group: $S_6$
Conductor: \(662908009\)\(\medspace = 25747^{2} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 6.0.25747.1
Galois orbit size: $1$
Smallest permutation container: 12T183
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $S_6$
Projective stem field: Galois closure of 6.0.25747.1

Defining polynomial

$f(x)$$=$ \( x^{6} - x^{5} + 2x^{4} - 2x^{3} + 3x^{2} - x + 1 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: \( x^{2} + 38x + 6 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 3 a + 14 + \left(20 a + 3\right)\cdot 41 + \left(13 a + 39\right)\cdot 41^{2} + \left(38 a + 40\right)\cdot 41^{3} + \left(16 a + 22\right)\cdot 41^{4} +O(41^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 18 + 27\cdot 41 + 29\cdot 41^{2} + 3\cdot 41^{3} + 32\cdot 41^{4} +O(41^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 35 a + \left(2 a + 30\right)\cdot 41 + \left(38 a + 35\right)\cdot 41^{2} + \left(40 a + 35\right)\cdot 41^{3} + \left(35 a + 25\right)\cdot 41^{4} +O(41^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 6 a + 23 + \left(38 a + 3\right)\cdot 41 + \left(2 a + 24\right)\cdot 41^{2} + 38\cdot 41^{3} + \left(5 a + 10\right)\cdot 41^{4} +O(41^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 5 + 39\cdot 41 + 16\cdot 41^{2} + 25\cdot 41^{3} + 36\cdot 41^{4} +O(41^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 38 a + 23 + \left(20 a + 19\right)\cdot 41 + \left(27 a + 18\right)\cdot 41^{2} + \left(2 a + 19\right)\cdot 41^{3} + \left(24 a + 35\right)\cdot 41^{4} +O(41^{5})\) 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$$1$$()$$5$
$15$$2$$(1,2)(3,4)(5,6)$$-3$
$15$$2$$(1,2)$$1$
$45$$2$$(1,2)(3,4)$$1$
$40$$3$$(1,2,3)(4,5,6)$$2$
$40$$3$$(1,2,3)$$-1$
$90$$4$$(1,2,3,4)(5,6)$$-1$
$90$$4$$(1,2,3,4)$$-1$
$144$$5$$(1,2,3,4,5)$$0$
$120$$6$$(1,2,3,4,5,6)$$0$
$120$$6$$(1,2,3)(4,5)$$1$

The blue line marks the conjugacy class containing complex conjugation.