Basic invariants
| Dimension: | $4$ |
| Group: | $S_5$ |
| Conductor: | \(4903\) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.3.4903.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_5$ |
| Parity: | odd |
| Determinant: | 1.4903.2t1.a.a |
| Projective image: | $S_5$ |
| Projective stem field: | Galois closure of 5.3.4903.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - x^{4} - x^{3} + 2x^{2} - x - 1 \)
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The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$:
\( x^{2} + 33x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 9 + 11\cdot 37 + 7\cdot 37^{2} + 37^{3} + 22\cdot 37^{4} +O(37^{5})\)
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| $r_{ 2 }$ | $=$ |
\( 20 + 8\cdot 37 + 24\cdot 37^{2} + 29\cdot 37^{3} + 37^{4} +O(37^{5})\)
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| $r_{ 3 }$ | $=$ |
\( 3 a + 26 + \left(6 a + 34\right)\cdot 37 + \left(23 a + 33\right)\cdot 37^{2} + \left(31 a + 22\right)\cdot 37^{3} + \left(13 a + 19\right)\cdot 37^{4} +O(37^{5})\)
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| $r_{ 4 }$ | $=$ |
\( 19 + 36\cdot 37^{2} + 4\cdot 37^{3} + 24\cdot 37^{4} +O(37^{5})\)
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| $r_{ 5 }$ | $=$ |
\( 34 a + 1 + \left(30 a + 19\right)\cdot 37 + \left(13 a + 9\right)\cdot 37^{2} + \left(5 a + 15\right)\cdot 37^{3} + \left(23 a + 6\right)\cdot 37^{4} +O(37^{5})\)
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Generators of the action on the roots $r_1, \ldots, r_{ 5 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 5 }$ | Character value | Complex conjugation |
| $1$ | $1$ | $()$ | $4$ | |
| $10$ | $2$ | $(1,2)$ | $2$ | ✓ |
| $15$ | $2$ | $(1,2)(3,4)$ | $0$ | |
| $20$ | $3$ | $(1,2,3)$ | $1$ | |
| $30$ | $4$ | $(1,2,3,4)$ | $0$ | |
| $24$ | $5$ | $(1,2,3,4,5)$ | $-1$ | |
| $20$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |