Basic invariants
| Dimension: | $3$ |
| Group: | $S_4$ |
| Conductor: | \(331\) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 4.2.331.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_4$ |
| Parity: | odd |
| Determinant: | 1.331.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.2.331.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} - x^{3} + x^{2} + x - 1 \)
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The roots of $f$ are computed in $\Q_{ 113 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 40 + 35\cdot 113 + 41\cdot 113^{2} + 79\cdot 113^{3} + 67\cdot 113^{4} +O(113^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 94 + 58\cdot 113 + 20\cdot 113^{2} + 65\cdot 113^{3} + 55\cdot 113^{4} +O(113^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 96 + 64\cdot 113 + 113^{2} + 99\cdot 113^{3} + 91\cdot 113^{4} +O(113^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 110 + 66\cdot 113 + 49\cdot 113^{2} + 95\cdot 113^{3} + 10\cdot 113^{4} +O(113^{5})\)
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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 | Complex conjugation |
| $1$ | $1$ | $()$ | $3$ | |
| $3$ | $2$ | $(1,2)(3,4)$ | $-1$ | |
| $6$ | $2$ | $(1,2)$ | $1$ | ✓ |
| $8$ | $3$ | $(1,2,3)$ | $0$ | |
| $6$ | $4$ | $(1,2,3,4)$ | $-1$ |