Properties

Label 5.646541456023.6t14.a.a
Dimension $5$
Group $S_5$
Conductor $646541456023$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $5$
Group: $S_5$
Conductor: \(646541456023\)\(\medspace = 8647^{3} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 5.3.8647.1
Galois orbit size: $1$
Smallest permutation container: $\PGL(2,5)$
Parity: odd
Determinant: 1.8647.2t1.a.a
Projective image: $S_5$
Projective stem field: Galois closure of 5.3.8647.1

Defining polynomial

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

The roots of $f$ are computed in $\Q_{ 563 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 33 + 208\cdot 563 + 332\cdot 563^{2} + 389\cdot 563^{3} + 74\cdot 563^{4} +O(563^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 360 + 91\cdot 563 + 108\cdot 563^{2} + 190\cdot 563^{3} + 235\cdot 563^{4} +O(563^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 363 + 532\cdot 563 + 519\cdot 563^{2} + 379\cdot 563^{3} + 19\cdot 563^{4} +O(563^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 434 + 492\cdot 563 + 53\cdot 563^{2} + 293\cdot 563^{3} + 409\cdot 563^{4} +O(563^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 499 + 363\cdot 563 + 111\cdot 563^{2} + 436\cdot 563^{3} + 386\cdot 563^{4} +O(563^{5})\) 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.