Properties

Label 3.9612.4t5.a.a
Dimension $3$
Group $S_4$
Conductor $9612$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $3$
Group: $S_4$
Conductor: \(9612\)\(\medspace = 2^{2} \cdot 3^{3} \cdot 89 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 4.2.9612.1
Galois orbit size: $1$
Smallest permutation container: $S_4$
Parity: odd
Determinant: 1.267.2t1.a.a
Projective image: $S_4$
Projective stem field: Galois closure of 4.2.9612.1

Defining polynomial

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

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

Roots:
$r_{ 1 }$ $=$ \( 70 + 39\cdot 137 + 127\cdot 137^{2} + 93\cdot 137^{3} + 17\cdot 137^{4} +O(137^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 83 + 70\cdot 137 + 32\cdot 137^{2} + 12\cdot 137^{3} + 92\cdot 137^{4} +O(137^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 126 + 104\cdot 137 + 34\cdot 137^{2} + 32\cdot 137^{3} + 14\cdot 137^{4} +O(137^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 133 + 58\cdot 137 + 79\cdot 137^{2} + 135\cdot 137^{3} + 12\cdot 137^{4} +O(137^{5})\) 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.