Properties

Label 6.194...513.20t30.a.a
Dimension $6$
Group $S_5$
Conductor $1.945\times 10^{15}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $6$
Group: $S_5$
Conductor: \(1944559427246513\)\(\medspace = 7^{3} \cdot 11^{3} \cdot 1621^{3} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 5.5.124817.1
Galois orbit size: $1$
Smallest permutation container: 20T30
Parity: even
Determinant: 1.124817.2t1.a.a
Projective image: $S_5$
Projective stem field: Galois closure of 5.5.124817.1

Defining polynomial

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

The roots of $f$ are computed in $\Q_{ 457 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 44 + 244\cdot 457 + 322\cdot 457^{2} + 104\cdot 457^{3} + 437\cdot 457^{4} +O(457^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 268 + 134\cdot 457 + 199\cdot 457^{2} + 181\cdot 457^{3} + 337\cdot 457^{4} +O(457^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 331 + 223\cdot 457 + 448\cdot 457^{2} + 121\cdot 457^{3} + 278\cdot 457^{4} +O(457^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 361 + 161\cdot 457 + 409\cdot 457^{2} + 40\cdot 457^{3} + 119\cdot 457^{4} +O(457^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 367 + 149\cdot 457 + 448\cdot 457^{2} + 7\cdot 457^{3} + 199\cdot 457^{4} +O(457^{5})\) 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.