Properties

Label 3.870124.4t5.a.a
Dimension $3$
Group $S_4$
Conductor $870124$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $3$
Group: $S_4$
Conductor: \(870124\)\(\medspace = 2^{2} \cdot 19 \cdot 107^{2} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 4.2.870124.1
Galois orbit size: $1$
Smallest permutation container: $S_4$
Parity: odd
Determinant: 1.19.2t1.a.a
Projective image: $S_4$
Projective stem field: Galois closure of 4.2.870124.1

Defining polynomial

$f(x)$$=$ \( x^{4} - x^{3} - 7x^{2} + 17x - 32 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 157 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 35 + 107\cdot 157 + 151\cdot 157^{2} + 33\cdot 157^{3} + 87\cdot 157^{4} +O(157^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 81 + 70\cdot 157 + 83\cdot 157^{2} + 131\cdot 157^{3} + 13\cdot 157^{4} +O(157^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 94 + 83\cdot 157 + 17\cdot 157^{2} + 91\cdot 157^{3} + 132\cdot 157^{4} +O(157^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 105 + 52\cdot 157 + 61\cdot 157^{2} + 57\cdot 157^{3} + 80\cdot 157^{4} +O(157^{5})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 4 }$ Character value
$1$$1$$()$$3$
$3$$2$$(1,2)(3,4)$$-1$
$6$$2$$(1,2)$$1$
$8$$3$$(1,2,3)$$0$
$6$$4$$(1,2,3,4)$$-1$

The blue line marks the conjugacy class containing complex conjugation.