Properties

Label 3.157609.4t4.a.a
Dimension $3$
Group $A_4$
Conductor $157609$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $3$
Group: $A_4$
Conductor: \(157609\)\(\medspace = 397^{2}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 4.4.157609.1
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 4.4.157609.1

Defining polynomial

$f(x)$$=$ \( x^{4} - 13x^{2} - 2x + 19 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 103 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 5 + 82\cdot 103 + 8\cdot 103^{2} + 21\cdot 103^{3} + 31\cdot 103^{4} +O(103^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 15 + 21\cdot 103 + 6\cdot 103^{2} + 53\cdot 103^{3} + 78\cdot 103^{4} +O(103^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 22 + 77\cdot 103 + 2\cdot 103^{2} + 19\cdot 103^{3} + 85\cdot 103^{4} +O(103^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 61 + 25\cdot 103 + 85\cdot 103^{2} + 9\cdot 103^{3} + 11\cdot 103^{4} +O(103^{5})\) 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.