Properties

Label 3.1489.4t5.a.a
Dimension $3$
Group $S_4$
Conductor $1489$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $3$
Group: $S_4$
Conductor: \(1489\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 4.0.1489.1
Galois orbit size: $1$
Smallest permutation container: $S_4$
Parity: even
Determinant: 1.1489.2t1.a.a
Projective image: $S_4$
Projective stem field: Galois closure of 4.0.1489.1

Defining polynomial

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

The roots of $f$ are computed in $\Q_{ 61 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 6 + 59\cdot 61 + 55\cdot 61^{2} + 3\cdot 61^{3} + 59\cdot 61^{4} +O(61^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 21 + 45\cdot 61 + 16\cdot 61^{2} + 42\cdot 61^{3} + 19\cdot 61^{4} +O(61^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 44 + 20\cdot 61 + 44\cdot 61^{2} + 18\cdot 61^{3} + 59\cdot 61^{4} +O(61^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 52 + 57\cdot 61 + 4\cdot 61^{2} + 57\cdot 61^{3} + 44\cdot 61^{4} +O(61^{5})\) 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 valueComplex conjugation
$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$