Properties

Label 5.4483962449.6t14.a.a
Dimension $5$
Group $S_5$
Conductor $4483962449$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $5$
Group: $S_5$
Conductor: \(4483962449\)\(\medspace = 17^{3} \cdot 97^{3} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 5.1.1649.1
Galois orbit size: $1$
Smallest permutation container: $\PGL(2,5)$
Parity: even
Determinant: 1.1649.2t1.a.a
Projective image: $S_5$
Projective stem field: Galois closure of 5.1.1649.1

Defining polynomial

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

The roots of $f$ are computed in $\Q_{ 499 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 121 + 359\cdot 499 + 431\cdot 499^{2} + 56\cdot 499^{3} + 376\cdot 499^{4} +O(499^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 123 + 266\cdot 499 + 309\cdot 499^{2} + 381\cdot 499^{3} + 84\cdot 499^{4} +O(499^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 163 + 125\cdot 499 + 219\cdot 499^{2} + 146\cdot 499^{3} + 173\cdot 499^{4} +O(499^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 180 + 69\cdot 499 + 489\cdot 499^{2} + 150\cdot 499^{3} +O(499^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 412 + 177\cdot 499 + 47\cdot 499^{2} + 262\cdot 499^{3} + 363\cdot 499^{4} +O(499^{5})\) 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 value
$1$$1$$()$$5$
$10$$2$$(1,2)$$-1$
$15$$2$$(1,2)(3,4)$$1$
$20$$3$$(1,2,3)$$-1$
$30$$4$$(1,2,3,4)$$1$
$24$$5$$(1,2,3,4,5)$$0$
$20$$6$$(1,2,3)(4,5)$$-1$

The blue line marks the conjugacy class containing complex conjugation.