Properties

Label 4.1609.5t5.1c1
Dimension 4
Group $S_5$
Conductor $ 1609 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$4$
Group:$S_5$
Conductor:$1609 $
Artin number field: Splitting field of $f=x^{5} - x^{3} - x^{2} + x + 1$ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: 5T5
Parity: Even
Determinant: \(\displaystyle\left(\frac{1609}{\bullet}\right)\)

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: \[ \begin{aligned} r_{ 1 } &= 9220153 +O\left(31^{ 5 }\right) \\ r_{ 2 } &= -13897449 a + 537036 +O\left(31^{ 5 }\right) \\ r_{ 3 } &= 13897449 a + 2754943 +O\left(31^{ 5 }\right) \\ r_{ 4 } &= -6558550 a + 13161563 +O\left(31^{ 5 }\right) \\ r_{ 5 } &= 6558550 a + 2955456 +O\left(31^{ 5 }\right) \\ \end{aligned}\]

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$$1$$()$$4$
$10$$2$$(1,2)$$2$
$15$$2$$(1,2)(3,4)$$0$
$20$$3$$(1,2,3)$$1$
$30$$4$$(1,2,3,4)$$0$
$24$$5$$(1,2,3,4,5)$$-1$
$20$$6$$(1,2,3)(4,5)$$-1$
The blue line marks the conjugacy class containing complex conjugation.