Properties

Label 4.53_139.5t5.1c1
Dimension 4
Group $S_5$
Conductor $ 53 \cdot 139 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$4$
Group:$S_5$
Conductor:$7367= 53 \cdot 139 $
Artin number field: Splitting field of $f= x^{5} - 2 x^{4} + 3 x^{2} - 2 x - 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_5$
Parity: Odd
Determinant: 1.53_139.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 277 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 5 + 154\cdot 277 + 271\cdot 277^{2} + 170\cdot 277^{3} + 206\cdot 277^{4} +O\left(277^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 145 + 54\cdot 277 + 47\cdot 277^{2} + 208\cdot 277^{3} + 90\cdot 277^{4} +O\left(277^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 184 + 269\cdot 277 + 242\cdot 277^{2} + 209\cdot 277^{3} + 31\cdot 277^{4} +O\left(277^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 228 + 192\cdot 277 + 114\cdot 277^{2} + 122\cdot 277^{3} + 272\cdot 277^{4} +O\left(277^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 271 + 159\cdot 277 + 154\cdot 277^{2} + 119\cdot 277^{3} + 229\cdot 277^{4} +O\left(277^{ 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.