Properties

Label 5.115...287.6t14.a.a
Dimension $5$
Group $S_5$
Conductor $1.152\times 10^{13}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $5$
Group: $S_5$
Conductor: \(11517146829287\)\(\medspace = 11^{3} \cdot 2053^{3}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 5.3.22583.1
Galois orbit size: $1$
Smallest permutation container: $\PGL(2,5)$
Parity: odd
Determinant: 1.22583.2t1.a.a
Projective image: $S_5$
Projective stem field: 5.3.22583.1

Defining polynomial

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

The roots of $f$ are computed in $\Q_{ 157 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 16 + 103\cdot 157 + 13\cdot 157^{2} + 151\cdot 157^{3} + 13\cdot 157^{4} +O(157^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 53 + 6\cdot 157 + 17\cdot 157^{2} + 135\cdot 157^{3} + 45\cdot 157^{4} +O(157^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 55 + 69\cdot 157 + 73\cdot 157^{2} + 21\cdot 157^{3} + 114\cdot 157^{4} +O(157^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 64 + 7\cdot 157 + 106\cdot 157^{2} + 21\cdot 157^{3} + 138\cdot 157^{4} +O(157^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 126 + 127\cdot 157 + 103\cdot 157^{2} + 141\cdot 157^{3} + 157^{4} +O(157^{5})\)  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.