Properties

Label 3.976.4t5.b.a
Dimension $3$
Group $S_4$
Conductor $976$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $3$
Group: $S_4$
Conductor: \(976\)\(\medspace = 2^{4} \cdot 61 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 4.2.976.1
Galois orbit size: $1$
Smallest permutation container: $S_4$
Parity: odd
Determinant: 1.244.2t1.a.a
Projective image: $S_4$
Projective stem field: Galois closure of 4.2.976.1

Defining polynomial

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

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

Roots:
$r_{ 1 }$ $=$ \( 10 + 14\cdot 43 + 25\cdot 43^{2} + 12\cdot 43^{3} + 25\cdot 43^{4} +O(43^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 11 + 5\cdot 43 + 21\cdot 43^{2} + 21\cdot 43^{3} + 28\cdot 43^{4} +O(43^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 26 + 35\cdot 43 + 20\cdot 43^{2} + 31\cdot 43^{3} + 37\cdot 43^{4} +O(43^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 41 + 30\cdot 43 + 18\cdot 43^{2} + 20\cdot 43^{3} + 37\cdot 43^{4} +O(43^{5})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 4 }$ Character valueComplex conjugation
$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$