Properties

Label 4.551...504.10t12.a.a
Dimension $4$
Group $S_5$
Conductor $5.516\times 10^{15}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $4$
Group: $S_5$
Conductor: \(5515586099981504\)\(\medspace = 2^{6} \cdot 44171^{3} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 5.5.176684.1
Galois orbit size: $1$
Smallest permutation container: $S_5$
Parity: even
Determinant: 1.176684.2t1.a.a
Projective image: $S_5$
Projective stem field: Galois closure of 5.5.176684.1

Defining polynomial

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

The roots of $f$ are computed in $\Q_{ 269 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 23 + 133\cdot 269 + 44\cdot 269^{2} + 33\cdot 269^{3} + 60\cdot 269^{4} +O(269^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 136 + 243\cdot 269 + 187\cdot 269^{2} + 260\cdot 269^{3} + 125\cdot 269^{4} +O(269^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 176 + 64\cdot 269 + 77\cdot 269^{2} + 41\cdot 269^{3} + 77\cdot 269^{4} +O(269^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 224 + 37\cdot 269 + 111\cdot 269^{2} + 8\cdot 269^{3} + 49\cdot 269^{4} +O(269^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 248 + 58\cdot 269 + 117\cdot 269^{2} + 194\cdot 269^{3} + 225\cdot 269^{4} +O(269^{5})\) 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$$1$$()$$4$
$10$$2$$(1,2)$$-2$
$15$$2$$(1,2)(3,4)$$0$
$20$$3$$(1,2,3)$$1$
$30$$4$$(1,2,3,4)$$0$
$24$$5$$(1,2,3,4,5)$$-1$
$20$$6$$(1,2,3)(4,5)$$1$

The blue line marks the conjugacy class containing complex conjugation.