Properties

Label 3.219024.4t4.a.a
Dimension $3$
Group $A_4$
Conductor $219024$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $3$
Group: $A_4$
Conductor: \(219024\)\(\medspace = 2^{4} \cdot 3^{4} \cdot 13^{2}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 4.0.219024.1
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 4.0.219024.1

Defining polynomial

$f(x)$$=$ \( x^{4} - 2x^{3} + 3x^{2} + 6x + 6 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 53 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 9 + 49\cdot 53 + 42\cdot 53^{2} + 4\cdot 53^{3} + 15\cdot 53^{4} +O(53^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 20 + 51\cdot 53 + 15\cdot 53^{2} + 12\cdot 53^{3} + 27\cdot 53^{4} +O(53^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 39 + 12\cdot 53 + 34\cdot 53^{3} + 9\cdot 53^{4} +O(53^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 40 + 45\cdot 53 + 46\cdot 53^{2} + 53^{3} + 53^{4} +O(53^{5})\) 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 value
$1$$1$$()$$3$
$3$$2$$(1,2)(3,4)$$-1$
$4$$3$$(1,2,3)$$0$
$4$$3$$(1,3,2)$$0$

The blue line marks the conjugacy class containing complex conjugation.