Properties

Label 6.299...329.20t30.a.a
Dimension $6$
Group $S_5$
Conductor $2.999\times 10^{15}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $6$
Group: $S_5$
Conductor: \(2999004351321329\)\(\medspace = 13^{3} \cdot 11093^{3} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 5.5.144209.1
Galois orbit size: $1$
Smallest permutation container: 20T30
Parity: even
Determinant: 1.144209.2t1.a.a
Projective image: $S_5$
Projective stem field: Galois closure of 5.5.144209.1

Defining polynomial

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

The roots of $f$ are computed in $\Q_{ 227 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 56 + 72\cdot 227 + 161\cdot 227^{2} + 128\cdot 227^{3} + 144\cdot 227^{4} +O(227^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 119 + 145\cdot 227 + 138\cdot 227^{2} + 152\cdot 227^{3} + 139\cdot 227^{4} +O(227^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 129 + 142\cdot 227 + 97\cdot 227^{2} + 103\cdot 227^{3} + 59\cdot 227^{4} +O(227^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 154 + 225\cdot 227 + 147\cdot 227^{2} + 225\cdot 227^{3} + 35\cdot 227^{4} +O(227^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 223 + 94\cdot 227 + 135\cdot 227^{2} + 70\cdot 227^{3} + 74\cdot 227^{4} +O(227^{5})\) 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$$1$$()$$6$
$10$$2$$(1,2)$$0$
$15$$2$$(1,2)(3,4)$$-2$
$20$$3$$(1,2,3)$$0$
$30$$4$$(1,2,3,4)$$0$
$24$$5$$(1,2,3,4,5)$$1$
$20$$6$$(1,2,3)(4,5)$$0$

The blue line marks the conjugacy class containing complex conjugation.