Properties

Label 5.41134735489.10t13.a.a
Dimension $5$
Group $S_5$
Conductor $41134735489$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $5$
Group: $S_5$
Conductor: \(41134735489\)\(\medspace = 202817^{2} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 5.5.202817.1
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 5.5.202817.1

Defining polynomial

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

The roots of $f$ are computed in $\Q_{ 347 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 6 + 27\cdot 347 + 187\cdot 347^{2} + 57\cdot 347^{3} + 183\cdot 347^{4} +O(347^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 131 + 215\cdot 347 + 18\cdot 347^{2} + 339\cdot 347^{3} + 234\cdot 347^{4} +O(347^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 178 + 36\cdot 347 + 130\cdot 347^{2} + 188\cdot 347^{3} + 279\cdot 347^{4} +O(347^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 190 + 168\cdot 347 + 193\cdot 347^{2} + 36\cdot 347^{3} + 18\cdot 347^{4} +O(347^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 191 + 246\cdot 347 + 164\cdot 347^{2} + 72\cdot 347^{3} + 325\cdot 347^{4} +O(347^{5})\) 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.