Properties

Label 3.6056521.6t8.a.a
Dimension $3$
Group $S_4$
Conductor $6056521$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $3$
Group: $S_4$
Conductor: \(6056521\)\(\medspace = 23^{2} \cdot 107^{2} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 4.2.263327.1
Galois orbit size: $1$
Smallest permutation container: $S_4$
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $S_4$
Projective stem field: Galois closure of 4.2.263327.1

Defining polynomial

$f(x)$$=$ \( x^{4} - x^{3} + 9x^{2} + 9x - 26 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 101 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 19 + 9\cdot 101 + 70\cdot 101^{2} + 2\cdot 101^{3} + 31\cdot 101^{4} +O(101^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 42 + 13\cdot 101 + 31\cdot 101^{2} + 86\cdot 101^{3} + 34\cdot 101^{4} +O(101^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 64 + 95\cdot 101 + 71\cdot 101^{2} + 46\cdot 101^{3} + 36\cdot 101^{4} +O(101^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 78 + 83\cdot 101 + 28\cdot 101^{2} + 66\cdot 101^{3} + 99\cdot 101^{4} +O(101^{5})\) Copy content Toggle raw display

Generators of the action on the roots $r_1, \ldots, r_{ 4 }$

Cycle notation
$(1,2,3,4)$
$(1,2)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 4 }$ Character value
$1$$1$$()$$3$
$3$$2$$(1,2)(3,4)$$-1$
$6$$2$$(1,2)$$-1$
$8$$3$$(1,2,3)$$0$
$6$$4$$(1,2,3,4)$$1$

The blue line marks the conjugacy class containing complex conjugation.