Properties

Label 4.4549.5t5.b.a
Dimension $4$
Group $S_5$
Conductor $4549$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $4$
Group: $S_5$
Conductor: \(4549\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 5.1.4549.1
Galois orbit size: $1$
Smallest permutation container: $S_5$
Parity: even
Determinant: 1.4549.2t1.a.a
Projective image: $S_5$
Projective stem field: Galois closure of 5.1.4549.1

Defining polynomial

$f(x)$$=$ \( x^{5} - x^{4} + 2x^{3} - 2x^{2} - 1 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 73 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 14 + 38\cdot 73 + 40\cdot 73^{2} + 39\cdot 73^{3} + 34\cdot 73^{4} +O(73^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 29 + 35\cdot 73 + 34\cdot 73^{2} + 41\cdot 73^{4} +O(73^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 43 + 44\cdot 73 + 5\cdot 73^{2} + 21\cdot 73^{3} + 66\cdot 73^{4} +O(73^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 63 + 64\cdot 73 + 25\cdot 73^{2} + 63\cdot 73^{3} + 46\cdot 73^{4} +O(73^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 71 + 35\cdot 73 + 39\cdot 73^{2} + 21\cdot 73^{3} + 30\cdot 73^{4} +O(73^{5})\) Copy content Toggle raw display

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 valueComplex conjugation
$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$