Properties

Label 3.480249.4t4.a.a
Dimension $3$
Group $A_4$
Conductor $480249$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $3$
Group: $A_4$
Conductor: \(480249\)\(\medspace = 3^{4} \cdot 7^{2} \cdot 11^{2} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 4.0.480249.1
Galois orbit size: $1$
Smallest permutation container: $A_4$
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $A_4$
Projective stem field: Galois closure of 4.0.480249.1

Defining polynomial

$f(x)$$=$ \( x^{4} - x^{3} - 3x^{2} - 30x + 75 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 131 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 24 + 81\cdot 131 + 5\cdot 131^{2} + 55\cdot 131^{3} + 73\cdot 131^{4} +O(131^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 38 + 61\cdot 131 + 30\cdot 131^{2} + 94\cdot 131^{3} + 2\cdot 131^{4} +O(131^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 77 + 75\cdot 131 + 76\cdot 131^{2} + 75\cdot 131^{3} + 91\cdot 131^{4} +O(131^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 124 + 43\cdot 131 + 18\cdot 131^{2} + 37\cdot 131^{3} + 94\cdot 131^{4} +O(131^{5})\) Copy content Toggle raw display

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

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

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$
$4$$3$$(1,2,3)$$0$
$4$$3$$(1,3,2)$$0$