Properties

Label 5.36497e3.6t14.1c1
Dimension 5
Group $S_5$
Conductor $ 36497^{3}$
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$5$
Group:$S_5$
Conductor:$48615135735473= 36497^{3} $
Artin number field: Splitting field of $f= x^{5} - 2 x^{4} - 3 x^{3} + 5 x^{2} + x - 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $\PGL(2,5)$
Parity: Even
Determinant: 1.36497.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: $ x^{2} + 29 x + 3 $
Roots: \[ \begin{aligned} r_{ 1 } &= 5 a + 3 + \left(4 a + 27\right)\cdot 31 + \left(11 a + 28\right)\cdot 31^{2} + \left(19 a + 28\right)\cdot 31^{3} + \left(11 a + 8\right)\cdot 31^{4} +O\left(31^{ 5 }\right) \\ r_{ 2 } &= 7 a + 15 + \left(18 a + 24\right)\cdot 31 + \left(12 a + 2\right)\cdot 31^{2} + \left(6 a + 1\right)\cdot 31^{3} + \left(28 a + 18\right)\cdot 31^{4} +O\left(31^{ 5 }\right) \\ r_{ 3 } &= 26 a + 13 + \left(26 a + 30\right)\cdot 31 + \left(19 a + 15\right)\cdot 31^{2} + \left(11 a + 25\right)\cdot 31^{3} + \left(19 a + 12\right)\cdot 31^{4} +O\left(31^{ 5 }\right) \\ r_{ 4 } &= 4 + 19\cdot 31 + 4\cdot 31^{2} + 5\cdot 31^{3} + 16\cdot 31^{4} +O\left(31^{ 5 }\right) \\ r_{ 5 } &= 24 a + 29 + \left(12 a + 22\right)\cdot 31 + \left(18 a + 9\right)\cdot 31^{2} + \left(24 a + 1\right)\cdot 31^{3} + \left(2 a + 6\right)\cdot 31^{4} +O\left(31^{ 5 }\right) \\ \end{aligned}\]

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.