Properties

Label 4.5437.5t5.1c1
Dimension 4
Group $S_5$
Conductor $ 5437 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$4$
Group:$S_5$
Conductor:$5437 $
Artin number field: Splitting field of $f= x^{5} - 2 x^{4} + 2 x^{3} - 2 x^{2} - 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_5$
Parity: Even
Determinant: 1.5437.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$: $ x^{2} + 42 x + 3 $
Roots:
$r_{ 1 }$ $=$ $ 28 + 40\cdot 43 + 38\cdot 43^{2} + 29\cdot 43^{3} + 14\cdot 43^{4} +O\left(43^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 24 a + 41 + \left(22 a + 6\right)\cdot 43 + \left(11 a + 39\right)\cdot 43^{2} + \left(14 a + 10\right)\cdot 43^{3} + \left(33 a + 1\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 22 + 4\cdot 43 + 12\cdot 43^{2} + 42\cdot 43^{3} +O\left(43^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 18 + 28\cdot 43 + 10\cdot 43^{2} + 32\cdot 43^{3} + 5\cdot 43^{4} +O\left(43^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 19 a + 22 + \left(20 a + 5\right)\cdot 43 + \left(31 a + 28\right)\cdot 43^{2} + \left(28 a + 13\right)\cdot 43^{3} + \left(9 a + 20\right)\cdot 43^{4} +O\left(43^{ 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.