Properties

Label 4.4597.5t5.1c1
Dimension 4
Group $S_5$
Conductor $ 4597 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$4$
Group:$S_5$
Conductor:$4597 $
Artin number field: Splitting field of $f= x^{5} + x^{3} - 2 x^{2} - x - 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_5$
Parity: Even
Determinant: 1.4597.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 353 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 113 + 146\cdot 353 + 304\cdot 353^{2} + 348\cdot 353^{3} + 275\cdot 353^{4} +O\left(353^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 152 + 112\cdot 353 + 231\cdot 353^{2} + 243\cdot 353^{3} + 275\cdot 353^{4} +O\left(353^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 164 + 221\cdot 353 + 253\cdot 353^{2} + 199\cdot 353^{3} + 171\cdot 353^{4} +O\left(353^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 292 + 197\cdot 353 + 331\cdot 353^{2} + 273\cdot 353^{3} + 3\cdot 353^{4} +O\left(353^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 338 + 27\cdot 353 + 291\cdot 353^{2} + 345\cdot 353^{3} + 331\cdot 353^{4} +O\left(353^{ 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.