Properties

Label 3.229.4t5.a
Dimension $3$
Group $S_4$
Conductor $229$
Indicator $1$

Related objects

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Basic invariants

Dimension:$3$
Group:$S_4$
Conductor:\(229\)
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
Projective image: $S_4$
Projective field: Galois closure of 4.0.229.1

Galois action

Roots of defining polynomial

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 values
$c1$
$1$ $1$ $()$ $3$
$3$ $2$ $(1,2)(3,4)$ $-1$
$6$ $2$ $(1,2)$ $1$
$8$ $3$ $(1,2,3)$ $0$
$6$ $4$ $(1,2,3,4)$ $-1$
The blue line marks the conjugacy class containing complex conjugation.