Properties

Label 5.54272689.10t13.a.a
Dimension $5$
Group $S_5$
Conductor $54272689$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $5$
Group: $S_5$
Conductor: \(54272689\)\(\medspace = 53^{2} \cdot 139^{2} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 5.3.7367.1
Galois orbit size: $1$
Smallest permutation container: $S_5$
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $S_5$
Projective stem field: Galois closure of 5.3.7367.1

Defining polynomial

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

The roots of $f$ are computed in $\Q_{ 277 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 5 + 154\cdot 277 + 271\cdot 277^{2} + 170\cdot 277^{3} + 206\cdot 277^{4} +O(277^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 145 + 54\cdot 277 + 47\cdot 277^{2} + 208\cdot 277^{3} + 90\cdot 277^{4} +O(277^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 184 + 269\cdot 277 + 242\cdot 277^{2} + 209\cdot 277^{3} + 31\cdot 277^{4} +O(277^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 228 + 192\cdot 277 + 114\cdot 277^{2} + 122\cdot 277^{3} + 272\cdot 277^{4} +O(277^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 271 + 159\cdot 277 + 154\cdot 277^{2} + 119\cdot 277^{3} + 229\cdot 277^{4} +O(277^{5})\) Copy content 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.