Properties

Label 3.52441.6t8.c.a
Dimension $3$
Group $S_4$
Conductor $52441$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $3$
Group: $S_4$
Conductor: \(52441\)\(\medspace = 229^{2}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 4.0.229.1
Galois orbit size: $1$
Smallest permutation container: $S_4$
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $S_4$
Projective field: Galois closure of 4.0.229.1

Defining polynomial

$f(x)$$=$$ x^{4} - x + 1 $.

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

Roots:
$r_{ 1 }$ $=$ $ 135 + 139\cdot 193 + 151\cdot 193^{2} + 188\cdot 193^{3} + 63\cdot 193^{4} +O\left(193^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 145 + 66\cdot 193 + 184\cdot 193^{2} + 103\cdot 193^{3} + 30\cdot 193^{4} +O\left(193^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 148 + 174\cdot 193 + 160\cdot 193^{2} + 29\cdot 193^{3} + 160\cdot 193^{4} +O\left(193^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 151 + 4\cdot 193 + 82\cdot 193^{2} + 63\cdot 193^{3} + 131\cdot 193^{4} +O\left(193^{ 5 }\right)$

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.