Basic invariants
| Dimension: | $4$ |
| Group: | $S_5$ |
| Conductor: | \(3369\)\(\medspace = 3 \cdot 1123 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.1.3369.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_5$ |
| Parity: | even |
| Determinant: | 1.3369.2t1.a.a |
| Projective image: | $S_5$ |
| Projective stem field: | Galois closure of 5.1.3369.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} + x^{3} - x^{2} - x - 1 \)
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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:
| $r_{ 1 }$ | $=$ |
\( 17 + 23\cdot 37 + 26\cdot 37^{2} + 28\cdot 37^{3} + 35\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 29 + 30\cdot 37 + 20\cdot 37^{2} + 8\cdot 37^{3} + 23\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 5 + 14\cdot 37 + 24\cdot 37^{2} + 3\cdot 37^{3} + 29\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 11 a + 8 + \left(3 a + 20\right)\cdot 37 + \left(5 a + 29\right)\cdot 37^{2} + \left(14 a + 27\right)\cdot 37^{3} + \left(28 a + 35\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 26 a + 15 + \left(33 a + 22\right)\cdot 37 + \left(31 a + 9\right)\cdot 37^{2} + \left(22 a + 5\right)\cdot 37^{3} + \left(8 a + 24\right)\cdot 37^{4} +O(37^{5})\)
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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 |
| $1$ | $1$ | $()$ | $4$ | |
| $10$ | $2$ | $(1,2)$ | $2$ | |
| $15$ | $2$ | $(1,2)(3,4)$ | $0$ | ✓ |
| $20$ | $3$ | $(1,2,3)$ | $1$ | |
| $30$ | $4$ | $(1,2,3,4)$ | $0$ | |
| $24$ | $5$ | $(1,2,3,4,5)$ | $-1$ | |
| $20$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |