Properties

Label 5.23339e3.6t14.1c1
Dimension 5
Group $S_5$
Conductor $ 23339^{3}$
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$5$
Group:$S_5$
Conductor:$12712961507219= 23339^{3} $
Artin number field: Splitting field of $f= x^{5} - x^{4} + 2 x^{2} - 2 x - 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $\PGL(2,5)$
Parity: Odd
Determinant: 1.23339.2t1.1c1

Galois action

Roots of defining polynomial

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\left(337^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 17 + 172\cdot 337 + 42\cdot 337^{2} + 18\cdot 337^{3} + 286\cdot 337^{4} +O\left(337^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 180 + 52\cdot 337 + 192\cdot 337^{2} + 200\cdot 337^{3} + 336\cdot 337^{4} +O\left(337^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 203 + 332\cdot 337 + 260\cdot 337^{2} + 4\cdot 337^{3} + 51\cdot 337^{4} +O\left(337^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 261 + 213\cdot 337 + 283\cdot 337^{2} + 215\cdot 337^{3} + 146\cdot 337^{4} +O\left(337^{ 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$$1$$()$$5$
$10$$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.