Properties

Label 5.33097009.10t13.a.a
Dimension $5$
Group $S_5$
Conductor $33097009$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $5$
Group: $S_5$
Conductor: \(33097009\)\(\medspace = 11^{2} \cdot 523^{2}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 5.1.5753.1
Galois orbit size: $1$
Smallest permutation container: $S_5$
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $S_5$
Projective stem field: 5.1.5753.1

Defining polynomial

$f(x)$$=$\(x^{5} - x^{4} - x^{3} + 2 x^{2} - x + 1\)  Toggle raw display.

The roots of $f$ are computed in $\Q_{ 467 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 103 + 452\cdot 467 + 304\cdot 467^{2} + 160\cdot 467^{3} + 237\cdot 467^{4} +O(467^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 209 + 205\cdot 467 + 99\cdot 467^{2} + 299\cdot 467^{3} + 120\cdot 467^{4} +O(467^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 256 + 414\cdot 467 + 251\cdot 467^{2} + 329\cdot 467^{3} + 236\cdot 467^{4} +O(467^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 397 + 338\cdot 467 + 360\cdot 467^{2} + 288\cdot 467^{3} + 198\cdot 467^{4} +O(467^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 437 + 456\cdot 467 + 383\cdot 467^{2} + 322\cdot 467^{3} + 140\cdot 467^{4} +O(467^{5})\)  Toggle raw display

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$$1$$()$$5$
$10$$2$$(1,2)$$1$
$15$$2$$(1,2)(3,4)$$1$
$20$$3$$(1,2,3)$$-1$
$30$$4$$(1,2,3,4)$$-1$
$24$$5$$(1,2,3,4,5)$$0$
$20$$6$$(1,2,3)(4,5)$$1$

The blue line marks the conjugacy class containing complex conjugation.