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: $S_5$
Parity: Even
Determinant: 1.1609.2t1.1c1

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: \[ \begin{aligned} r_{ 1 } &= 9 + 10\cdot 31 + 15\cdot 31^{2} + 30\cdot 31^{3} + 9\cdot 31^{4} +O\left(31^{ 5 }\right) \\ r_{ 2 } &= 6 a + 23 + \left(17 a + 25\right)\cdot 31 + 15 a\cdot 31^{2} + \left(29 a + 18\right)\cdot 31^{3} + 15 a\cdot 31^{4} +O\left(31^{ 5 }\right) \\ r_{ 3 } &= 25 a + 4 + \left(13 a + 23\right)\cdot 31 + \left(15 a + 14\right)\cdot 31^{2} + \left(a + 30\right)\cdot 31^{3} + \left(15 a + 2\right)\cdot 31^{4} +O\left(31^{ 5 }\right) \\ r_{ 4 } &= 27 a + 17 + \left(8 a + 21\right)\cdot 31 + \left(26 a + 24\right)\cdot 31^{2} + \left(27 a + 7\right)\cdot 31^{3} + \left(23 a + 14\right)\cdot 31^{4} +O\left(31^{ 5 }\right) \\ r_{ 5 } &= 4 a + 9 + \left(22 a + 12\right)\cdot 31 + \left(4 a + 6\right)\cdot 31^{2} + \left(3 a + 6\right)\cdot 31^{3} + \left(7 a + 3\right)\cdot 31^{4} +O\left(31^{ 5 }\right) \\ \end{aligned}\]

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$$1$$()$$4$
$10$$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.