Properties

Label 3.1791.4t5.b
Dimension $3$
Group $S_4$
Conductor $1791$
Indicator $1$

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Basic invariants

Dimension:$3$
Group:$S_4$
Conductor:\(1791\)\(\medspace = 3^{2} \cdot 199 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 4.2.1791.1
Galois orbit size: $1$
Smallest permutation container: $S_4$
Parity: odd
Projective image: $S_4$
Projective field: Galois closure of 4.2.1791.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 313 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ \( 166 + 201\cdot 313 + 6\cdot 313^{2} + 25\cdot 313^{3} + 242\cdot 313^{4} +O(313^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 196 + 298\cdot 313 + 173\cdot 313^{2} + 281\cdot 313^{3} + 231\cdot 313^{4} +O(313^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 269 + 292\cdot 313 + 37\cdot 313^{2} + 307\cdot 313^{3} + 247\cdot 313^{4} +O(313^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 309 + 145\cdot 313 + 94\cdot 313^{2} + 12\cdot 313^{3} + 217\cdot 313^{4} +O(313^{5})\) 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.