Properties

Label 3.1366561.4t4.a.a
Dimension $3$
Group $A_4$
Conductor $1366561$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $3$
Group: $A_4$
Conductor: \(1366561\)\(\medspace = 7^{2} \cdot 167^{2} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 4.4.1366561.1
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 4.4.1366561.1

Defining polynomial

$f(x)$$=$ \( x^{4} - x^{3} - 28x^{2} + 35x + 56 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 43 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 11 + 33\cdot 43 + 13\cdot 43^{2} + 24\cdot 43^{3} + 14\cdot 43^{4} +O(43^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 20 + 31\cdot 43 + 11\cdot 43^{2} + 27\cdot 43^{3} + 20\cdot 43^{4} +O(43^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 24 + 7\cdot 43 + 34\cdot 43^{2} + 23\cdot 43^{3} + 22\cdot 43^{4} +O(43^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 32 + 13\cdot 43 + 26\cdot 43^{2} + 10\cdot 43^{3} + 28\cdot 43^{4} +O(43^{5})\) 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.