Properties

Label 5.2682825616.10t13.a.a
Dimension $5$
Group $S_5$
Conductor $2682825616$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $5$
Group: $S_5$
Conductor: \(2682825616\)\(\medspace = 2^{4} \cdot 23^{2} \cdot 563^{2} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 5.5.207184.1
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 5.5.207184.1

Defining polynomial

$f(x)$$=$ \( x^{5} - x^{4} - 6x^{3} + 2x^{2} + 7x - 1 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 283 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 13 + 163\cdot 283 + 163\cdot 283^{2} + 114\cdot 283^{3} + 118\cdot 283^{4} +O(283^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 16 + 43\cdot 283 + 34\cdot 283^{2} + 11\cdot 283^{3} + 28\cdot 283^{4} +O(283^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 57 + 189\cdot 283 + 126\cdot 283^{2} + 169\cdot 283^{3} + 185\cdot 283^{4} +O(283^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 234 + 201\cdot 283 + 246\cdot 283^{2} + 142\cdot 283^{3} + 31\cdot 283^{4} +O(283^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 247 + 251\cdot 283 + 277\cdot 283^{2} + 127\cdot 283^{3} + 202\cdot 283^{4} +O(283^{5})\) 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.