Properties

Label 4.5501.5t5.1c1
Dimension 4
Group $S_5$
Conductor $ 5501 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$4$
Group:$S_5$
Conductor:$5501 $
Artin number field: Splitting field of $f= x^{5} - 2 x^{4} + 2 x^{2} - x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_5$
Parity: Even
Determinant: 1.5501.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: $ x^{2} + 45 x + 5 $
Roots:
$r_{ 1 }$ $=$ $ 25 + 19\cdot 47 + 46\cdot 47^{2} + 37\cdot 47^{3} + 23\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 9 a + 14 + \left(41 a + 8\right)\cdot 47 + \left(38 a + 2\right)\cdot 47^{2} + \left(44 a + 41\right)\cdot 47^{3} + \left(36 a + 3\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 20 a + 16 + \left(45 a + 27\right)\cdot 47 + \left(40 a + 8\right)\cdot 47^{2} + \left(a + 27\right)\cdot 47^{3} + 17 a\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 27 a + 9 + \left(a + 4\right)\cdot 47 + \left(6 a + 45\right)\cdot 47^{2} + \left(45 a + 36\right)\cdot 47^{3} + \left(29 a + 32\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 38 a + 32 + \left(5 a + 34\right)\cdot 47 + \left(8 a + 38\right)\cdot 47^{2} + \left(2 a + 44\right)\cdot 47^{3} + \left(10 a + 32\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$

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.