Properties

Label 4.4549.5t5.1c1
Dimension 4
Group $S_5$
Conductor $ 4549 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$4$
Group:$S_5$
Conductor:$4549 $
Artin number field: Splitting field of $f= x^{5} - 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: 1.4549.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 73 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 14 + 38\cdot 73 + 40\cdot 73^{2} + 39\cdot 73^{3} + 34\cdot 73^{4} +O\left(73^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 29 + 35\cdot 73 + 34\cdot 73^{2} + 41\cdot 73^{4} +O\left(73^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 43 + 44\cdot 73 + 5\cdot 73^{2} + 21\cdot 73^{3} + 66\cdot 73^{4} +O\left(73^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 63 + 64\cdot 73 + 25\cdot 73^{2} + 63\cdot 73^{3} + 46\cdot 73^{4} +O\left(73^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 71 + 35\cdot 73 + 39\cdot 73^{2} + 21\cdot 73^{3} + 30\cdot 73^{4} +O\left(73^{ 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.