Properties

Label 5.201642412544.6t14.a.a
Dimension $5$
Group $S_5$
Conductor $201642412544$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $5$
Group: $S_5$
Conductor: \(201642412544\)\(\medspace = 2^{9} \cdot 733^{3}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 5.1.5864.1
Galois orbit size: $1$
Smallest permutation container: $\PGL(2,5)$
Parity: even
Determinant: 1.5864.2t1.b.a
Projective image: $S_5$
Projective stem field: 5.1.5864.1

Defining polynomial

$f(x)$$=$\(x^{5} - x^{4} + x^{3} + 2 x^{2} - 2 x + 1\)  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 $\mathbb{Q}_{ 47 }$: \(x^{2} + 45 x + 5\)  Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 21 a + 16 + \left(35 a + 6\right)\cdot 47 + 41\cdot 47^{2} + \left(a + 24\right)\cdot 47^{3} + \left(8 a + 19\right)\cdot 47^{4} +O(47^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 21 a + 18 + \left(25 a + 25\right)\cdot 47 + \left(33 a + 23\right)\cdot 47^{2} + \left(18 a + 7\right)\cdot 47^{3} + \left(37 a + 26\right)\cdot 47^{4} +O(47^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 26 a + 11 + \left(11 a + 9\right)\cdot 47 + \left(46 a + 7\right)\cdot 47^{2} + \left(45 a + 26\right)\cdot 47^{3} + \left(38 a + 34\right)\cdot 47^{4} +O(47^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 26 a + 13 + \left(21 a + 8\right)\cdot 47 + \left(13 a + 18\right)\cdot 47^{2} + \left(28 a + 11\right)\cdot 47^{3} + \left(9 a + 35\right)\cdot 47^{4} +O(47^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 37 + 44\cdot 47 + 3\cdot 47^{2} + 24\cdot 47^{3} + 25\cdot 47^{4} +O(47^{5})\)  Toggle raw display

Generators of the action on the roots $r_1, \ldots, r_{ 5 }$

Cycle notation
$(1,2)$
$(1,2,3,4,5)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 5 }$ Character value
$1$$1$$()$$5$
$10$$2$$(1,2)$$-1$
$15$$2$$(1,2)(3,4)$$1$
$20$$3$$(1,2,3)$$-1$
$30$$4$$(1,2,3,4)$$1$
$24$$5$$(1,2,3,4,5)$$0$
$20$$6$$(1,2,3)(4,5)$$-1$

The blue line marks the conjugacy class containing complex conjugation.