Properties

Label 3.9801.6t8.e.a
Dimension $3$
Group $S_4$
Conductor $9801$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $3$
Group: $S_4$
Conductor: \(9801\)\(\medspace = 3^{4} \cdot 11^{2} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 4.2.3267.1
Galois orbit size: $1$
Smallest permutation container: $S_4$
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $S_4$
Projective stem field: Galois closure of 4.2.3267.1

Defining polynomial

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

The roots of $f$ are computed in $\Q_{ 193 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 5 + 113\cdot 193 + 4\cdot 193^{2} + 59\cdot 193^{3} + 153\cdot 193^{4} +O(193^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 43 + 138\cdot 193 + 174\cdot 193^{2} + 118\cdot 193^{3} + 143\cdot 193^{4} +O(193^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 71 + 20\cdot 193 + 120\cdot 193^{2} + 167\cdot 193^{3} + 65\cdot 193^{4} +O(193^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 75 + 114\cdot 193 + 86\cdot 193^{2} + 40\cdot 193^{3} + 23\cdot 193^{4} +O(193^{5})\) 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.