Properties

Label 5.221741881.10t13.a.a
Dimension $5$
Group $S_5$
Conductor $221741881$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $5$
Group: $S_5$
Conductor: \(221741881\)\(\medspace = 14891^{2}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 5.3.14891.1
Galois orbit size: $1$
Smallest permutation container: $S_5$
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $S_5$
Projective stem field: 5.3.14891.1

Defining polynomial

$f(x)$$=$\(x^{5} - x^{4} - 3 x^{3} + 3 x^{2} + x - 2\)  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 $\mathbb{Q}_{ 17 }$: \(x^{2} + 16 x + 3\)  Toggle raw display

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