Properties

Label 4.202817.5t5.a.a
Dimension 4
Group $S_5$
Conductor $ 202817 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$4$
Group:$S_5$
Conductor:$202817 $
Artin number field: Splitting field of 5.5.202817.1 defined by $f= x^{5} - 2 x^{4} - 4 x^{3} + 5 x^{2} + 2 x - 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_5$
Parity: Even
Determinant: 1.202817.2t1.a.a
Projective image: $S_5$
Projective field: Galois closure of 5.5.202817.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 347 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 6 + 27\cdot 347 + 187\cdot 347^{2} + 57\cdot 347^{3} + 183\cdot 347^{4} +O\left(347^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 131 + 215\cdot 347 + 18\cdot 347^{2} + 339\cdot 347^{3} + 234\cdot 347^{4} +O\left(347^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 178 + 36\cdot 347 + 130\cdot 347^{2} + 188\cdot 347^{3} + 279\cdot 347^{4} +O\left(347^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 190 + 168\cdot 347 + 193\cdot 347^{2} + 36\cdot 347^{3} + 18\cdot 347^{4} +O\left(347^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 191 + 246\cdot 347 + 164\cdot 347^{2} + 72\cdot 347^{3} + 325\cdot 347^{4} +O\left(347^{ 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.