Properties

Label 5.32171.6t16.a.a
Dimension $5$
Group $S_6$
Conductor $32171$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $5$
Group: $S_6$
Conductor: \(32171\)\(\medspace = 53 \cdot 607 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 6.0.32171.1
Galois orbit size: $1$
Smallest permutation container: $S_6$
Parity: odd
Determinant: 1.32171.2t1.a.a
Projective image: $S_6$
Projective stem field: Galois closure of 6.0.32171.1

Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: \( x^{2} + 45x + 5 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 20 a + 33 + \left(8 a + 9\right)\cdot 47 + \left(21 a + 38\right)\cdot 47^{2} + 9\cdot 47^{3} + \left(24 a + 21\right)\cdot 47^{4} +O(47^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 17 a + 41 + \left(42 a + 29\right)\cdot 47 + \left(5 a + 43\right)\cdot 47^{2} + \left(7 a + 42\right)\cdot 47^{3} + \left(a + 7\right)\cdot 47^{4} +O(47^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 27 a + 26 + \left(38 a + 6\right)\cdot 47 + \left(25 a + 25\right)\cdot 47^{2} + \left(46 a + 36\right)\cdot 47^{3} + \left(22 a + 21\right)\cdot 47^{4} +O(47^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 16 + 3\cdot 47 + 39\cdot 47^{2} + 17\cdot 47^{3} + 31\cdot 47^{4} +O(47^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 30 a + 28 + \left(4 a + 3\right)\cdot 47 + \left(41 a + 13\right)\cdot 47^{2} + \left(39 a + 4\right)\cdot 47^{3} + \left(45 a + 3\right)\cdot 47^{4} +O(47^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 45 + 40\cdot 47 + 28\cdot 47^{2} + 29\cdot 47^{3} + 8\cdot 47^{4} +O(47^{5})\) 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$$(1,2)(3,4)(5,6)$$-1$
$15$$2$$(1,2)$$3$
$45$$2$$(1,2)(3,4)$$1$
$40$$3$$(1,2,3)(4,5,6)$$-1$
$40$$3$$(1,2,3)$$2$
$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$$(1,2,3,4,5,6)$$-1$
$120$$6$$(1,2,3)(4,5)$$0$

The blue line marks the conjugacy class containing complex conjugation.