Properties

Label 5.138...637.6t14.a.a
Dimension $5$
Group $S_5$
Conductor $1.385\times 10^{16}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $5$
Group: $S_5$
Conductor: \(13846995138432637\)\(\medspace = 439^{3} \cdot 547^{3} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 5.5.240133.1
Galois orbit size: $1$
Smallest permutation container: $\PGL(2,5)$
Parity: even
Determinant: 1.240133.2t1.a.a
Projective image: $S_5$
Projective stem field: Galois closure of 5.5.240133.1

Defining polynomial

$f(x)$$=$ \( x^{5} - 2x^{4} - 4x^{3} + 6x^{2} + 2x - 1 \) Copy content 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 $\mathbb{Q}_{ 17 }$: \( x^{2} + 16x + 3 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 8 a + 6 + \left(13 a + 3\right)\cdot 17 + \left(7 a + 14\right)\cdot 17^{2} + \left(7 a + 4\right)\cdot 17^{3} + \left(5 a + 9\right)\cdot 17^{4} +O(17^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 10 + 5\cdot 17 + 5\cdot 17^{2} + 4\cdot 17^{3} + 3\cdot 17^{4} +O(17^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 15 a + 4 + \left(6 a + 12\right)\cdot 17 + 14\cdot 17^{2} + \left(3 a + 8\right)\cdot 17^{3} + \left(6 a + 5\right)\cdot 17^{4} +O(17^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 9 a + 14 + \left(3 a + 8\right)\cdot 17 + \left(9 a + 8\right)\cdot 17^{2} + \left(9 a + 4\right)\cdot 17^{3} + \left(11 a + 7\right)\cdot 17^{4} +O(17^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 2 a + 2 + \left(10 a + 4\right)\cdot 17 + \left(16 a + 8\right)\cdot 17^{2} + \left(13 a + 11\right)\cdot 17^{3} + \left(10 a + 8\right)\cdot 17^{4} +O(17^{5})\) Copy content 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.