Properties

Label 4.4903.5t5.1c1
Dimension 4
Group $S_5$
Conductor $ 4903 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$4$
Group:$S_5$
Conductor:$4903 $
Artin number field: Splitting field of $f= x^{5} - x^{4} - x^{3} + 2 x^{2} - x - 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_5$
Parity: Odd
Determinant: 1.4903.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: $ x^{2} + 33 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 9 + 11\cdot 37 + 7\cdot 37^{2} + 37^{3} + 22\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 20 + 8\cdot 37 + 24\cdot 37^{2} + 29\cdot 37^{3} + 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 3 a + 26 + \left(6 a + 34\right)\cdot 37 + \left(23 a + 33\right)\cdot 37^{2} + \left(31 a + 22\right)\cdot 37^{3} + \left(13 a + 19\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 19 + 36\cdot 37^{2} + 4\cdot 37^{3} + 24\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 34 a + 1 + \left(30 a + 19\right)\cdot 37 + \left(13 a + 9\right)\cdot 37^{2} + \left(5 a + 15\right)\cdot 37^{3} + \left(23 a + 6\right)\cdot 37^{4} +O\left(37^{ 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.