Properties

Label 3.61504.4t4.b.a
Dimension $3$
Group $A_4$
Conductor $61504$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $3$
Group: $A_4$
Conductor: \(61504\)\(\medspace = 2^{6} \cdot 31^{2}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 4.0.61504.1
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 4.0.61504.1

Defining polynomial

$f(x)$$=$ \( x^{4} - 6x^{2} - 8x + 28 \) Copy content Toggle raw display .

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

Roots:
$r_{ 1 }$ $=$ \( 17 + 48\cdot 61 + 21\cdot 61^{2} + 31\cdot 61^{3} + 3\cdot 61^{4} +O(61^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 28 + 48\cdot 61 + 41\cdot 61^{2} + 36\cdot 61^{3} + 30\cdot 61^{4} +O(61^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 38 + 18\cdot 61 + 19\cdot 61^{2} + 59\cdot 61^{3} + 32\cdot 61^{4} +O(61^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 39 + 6\cdot 61 + 39\cdot 61^{2} + 55\cdot 61^{3} + 54\cdot 61^{4} +O(61^{5})\) Copy content Toggle raw display

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

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

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.