Basic invariants
| Dimension: | $3$ |
| Group: | $S_4$ |
| Conductor: | \(751\) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 4.2.751.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_4$ |
| Parity: | odd |
| Determinant: | 1.751.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.2.751.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} - 2x^{3} + x^{2} - x - 1 \)
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The roots of $f$ are computed in $\Q_{ 163 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 32 + 98\cdot 163 + 133\cdot 163^{2} + 94\cdot 163^{3} + 70\cdot 163^{4} +O(163^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 34 + 7\cdot 163 + 75\cdot 163^{2} + 5\cdot 163^{3} + 99\cdot 163^{4} +O(163^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 118 + 112\cdot 163 + 142\cdot 163^{2} + 115\cdot 163^{3} + 75\cdot 163^{4} +O(163^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 144 + 107\cdot 163 + 137\cdot 163^{2} + 109\cdot 163^{3} + 80\cdot 163^{4} +O(163^{5})\)
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Generators of the action on the roots $r_1, \ldots, r_{ 4 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action 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.