Properties

Label 3.316969.6t8.b.a
Dimension $3$
Group $S_4$
Conductor $316969$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $3$
Group: $S_4$
Conductor: \(316969\)\(\medspace = 563^{2}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 4.2.563.1
Galois orbit size: $1$
Smallest permutation container: $S_4$
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $S_4$
Projective stem field: 4.2.563.1

Defining polynomial

$f(x)$$=$\(x^{4} - x^{3} + x^{2} - x - 1\)  Toggle raw display.

The roots of $f$ are computed in $\Q_{ 137 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 80 + 67\cdot 137 + 75\cdot 137^{2} + 127\cdot 137^{3} + 122\cdot 137^{4} +O(137^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 108 + 29\cdot 137 + 103\cdot 137^{2} + 73\cdot 137^{3} + 31\cdot 137^{4} +O(137^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 111 + 67\cdot 137 + 30\cdot 137^{2} + 52\cdot 137^{3} + 26\cdot 137^{4} +O(137^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 113 + 108\cdot 137 + 64\cdot 137^{2} + 20\cdot 137^{3} + 93\cdot 137^{4} +O(137^{5})\)  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.