Basic invariants
| Dimension: | $3$ |
| Group: | $S_4$ |
| Conductor: | \(50432\)\(\medspace = 2^{8} \cdot 197 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 4.0.50432.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_4$ |
| Parity: | even |
| Determinant: | 1.197.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.0.50432.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} + 4x^{2} - 4x + 6 \)
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The roots of $f$ are computed in an extension of $\Q_{ 5 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 5 }$:
\( x^{2} + 4x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 2 + 5 + 3\cdot 5^{2} + 2\cdot 5^{4} +O(5^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 4 + 2\cdot 5 + 5^{2} + 5^{4} +O(5^{5})\)
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| $r_{ 3 }$ | $=$ |
\( 4 a + 2\cdot 5 + \left(a + 2\right)\cdot 5^{2} + \left(3 a + 3\right)\cdot 5^{3} + \left(2 a + 3\right)\cdot 5^{4} +O(5^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( a + 4 + \left(4 a + 3\right)\cdot 5 + \left(3 a + 2\right)\cdot 5^{2} + a\cdot 5^{3} + \left(2 a + 3\right)\cdot 5^{4} +O(5^{5})\)
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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 |
| $1$ | $1$ | $()$ | $3$ | |
| $3$ | $2$ | $(1,2)(3,4)$ | $-1$ | ✓ |
| $6$ | $2$ | $(1,2)$ | $1$ | |
| $8$ | $3$ | $(1,2,3)$ | $0$ | |
| $6$ | $4$ | $(1,2,3,4)$ | $-1$ |