Properties

Label 6.127...219.20t30.a.a
Dimension $6$
Group $S_5$
Conductor $1.271\times 10^{13}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $6$
Group: $S_5$
Conductor: \(12712961507219\)\(\medspace = 23339^{3} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 5.3.23339.1
Galois orbit size: $1$
Smallest permutation container: 20T30
Parity: odd
Determinant: 1.23339.2t1.a.a
Projective image: $S_5$
Projective stem field: Galois closure of 5.3.23339.1

Defining polynomial

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

The roots of $f$ are computed in $\Q_{ 337 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 14 + 240\cdot 337 + 231\cdot 337^{2} + 234\cdot 337^{3} + 190\cdot 337^{4} +O(337^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 17 + 172\cdot 337 + 42\cdot 337^{2} + 18\cdot 337^{3} + 286\cdot 337^{4} +O(337^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 180 + 52\cdot 337 + 192\cdot 337^{2} + 200\cdot 337^{3} + 336\cdot 337^{4} +O(337^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 203 + 332\cdot 337 + 260\cdot 337^{2} + 4\cdot 337^{3} + 51\cdot 337^{4} +O(337^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 261 + 213\cdot 337 + 283\cdot 337^{2} + 215\cdot 337^{3} + 146\cdot 337^{4} +O(337^{5})\) 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$$1$$()$$6$
$10$$2$$(1,2)$$0$
$15$$2$$(1,2)(3,4)$$-2$
$20$$3$$(1,2,3)$$0$
$30$$4$$(1,2,3,4)$$0$
$24$$5$$(1,2,3,4,5)$$1$
$20$$6$$(1,2,3)(4,5)$$0$

The blue line marks the conjugacy class containing complex conjugation.