Basic invariants
| Dimension: | $4$ |
| Group: | $S_5$ |
| Conductor: | \(6686110323719\)\(\medspace = 18839^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.3.18839.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_5$ |
| Parity: | odd |
| Determinant: | 1.18839.2t1.a.a |
| Projective image: | $S_5$ |
| Projective stem field: | Galois closure of 5.3.18839.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - x^{4} + 3x^{2} - x - 1 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 25 a + 1 + \left(24 a + 5\right)\cdot 37 + \left(a + 22\right)\cdot 37^{2} + \left(20 a + 12\right)\cdot 37^{3} + \left(12 a + 4\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 30 + 12\cdot 37 + 15\cdot 37^{2} + 19\cdot 37^{3} + 27\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 12 a + 27 + \left(12 a + 4\right)\cdot 37 + \left(35 a + 4\right)\cdot 37^{2} + \left(16 a + 17\right)\cdot 37^{3} + \left(24 a + 34\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 3 + 10\cdot 37 + 14\cdot 37^{2} + 13\cdot 37^{3} + 10\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 14 + 4\cdot 37 + 18\cdot 37^{2} + 11\cdot 37^{3} + 34\cdot 37^{4} +O(37^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $4$ |
| $10$ | $2$ | $(1,2)$ | $-2$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $20$ | $3$ | $(1,2,3)$ | $1$ |
| $30$ | $4$ | $(1,2,3,4)$ | $0$ |
| $24$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
| $20$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.