Properties

Label 3.263327.4t5.a
Dimension $3$
Group $S_4$
Conductor $263327$
Indicator $1$

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Basic invariants

Dimension:$3$
Group:$S_4$
Conductor:\(263327\)\(\medspace = 23 \cdot 107^{2} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 4.2.263327.1
Galois orbit size: $1$
Smallest permutation container: $S_4$
Parity: odd
Projective image: $S_4$
Projective field: Galois closure of 4.2.263327.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 101 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ \( 19 + 9\cdot 101 + 70\cdot 101^{2} + 2\cdot 101^{3} + 31\cdot 101^{4} +O(101^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 42 + 13\cdot 101 + 31\cdot 101^{2} + 86\cdot 101^{3} + 34\cdot 101^{4} +O(101^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 64 + 95\cdot 101 + 71\cdot 101^{2} + 46\cdot 101^{3} + 36\cdot 101^{4} +O(101^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 78 + 83\cdot 101 + 28\cdot 101^{2} + 66\cdot 101^{3} + 99\cdot 101^{4} +O(101^{5})\) 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 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.