Properties

Label 4.11_523.5t5.1c1
Dimension 4
Group $S_5$
Conductor $ 11 \cdot 523 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$4$
Group:$S_5$
Conductor:$5753= 11 \cdot 523 $
Artin number field: Splitting field of $f= x^{5} - x^{4} - x^{3} + 2 x^{2} - x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_5$
Parity: Even
Determinant: 1.11_523.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 467 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 103 + 452\cdot 467 + 304\cdot 467^{2} + 160\cdot 467^{3} + 237\cdot 467^{4} +O\left(467^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 209 + 205\cdot 467 + 99\cdot 467^{2} + 299\cdot 467^{3} + 120\cdot 467^{4} +O\left(467^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 256 + 414\cdot 467 + 251\cdot 467^{2} + 329\cdot 467^{3} + 236\cdot 467^{4} +O\left(467^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 397 + 338\cdot 467 + 360\cdot 467^{2} + 288\cdot 467^{3} + 198\cdot 467^{4} +O\left(467^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 437 + 456\cdot 467 + 383\cdot 467^{2} + 322\cdot 467^{3} + 140\cdot 467^{4} +O\left(467^{ 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.