Properties

Label 4.2e4_13997.5t5.1c1
Dimension 4
Group $S_5$
Conductor $ 2^{4} \cdot 13997 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$4$
Group:$S_5$
Conductor:$223952= 2^{4} \cdot 13997 $
Artin number field: Splitting field of $f= x^{5} - 6 x^{3} - 2 x^{2} + 6 x + 2 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_5$
Parity: Even
Determinant: 1.13997.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 173 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 26 + 146\cdot 173 + 116\cdot 173^{2} + 98\cdot 173^{3} + 84\cdot 173^{4} +O\left(173^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 53 + 48\cdot 173 + 14\cdot 173^{2} + 25\cdot 173^{3} + 164\cdot 173^{4} +O\left(173^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 58 + 145\cdot 173 + 138\cdot 173^{2} + 122\cdot 173^{3} + 44\cdot 173^{4} +O\left(173^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 100 + 94\cdot 173 + 35\cdot 173^{2} + 110\cdot 173^{3} + 49\cdot 173^{4} +O\left(173^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 109 + 84\cdot 173 + 40\cdot 173^{2} + 162\cdot 173^{3} + 2\cdot 173^{4} +O\left(173^{ 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$$1$$()$$4$
$10$$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$
The blue line marks the conjugacy class containing complex conjugation.