Basic invariants
| Dimension: | $4$ |
| Group: | $S_5$ |
| Conductor: | \(4165509529\)\(\medspace = 1609^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.1.1609.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_5$ |
| Parity: | even |
| Determinant: | 1.1609.2t1.a.a |
| Projective image: | $S_5$ |
| Projective stem field: | Galois closure of 5.1.1609.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - x^{3} - x^{2} + x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$:
\( x^{2} + 29x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 9 + 10\cdot 31 + 15\cdot 31^{2} + 30\cdot 31^{3} + 9\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 6 a + 23 + \left(17 a + 25\right)\cdot 31 + 15 a\cdot 31^{2} + \left(29 a + 18\right)\cdot 31^{3} + 15 a\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 25 a + 4 + \left(13 a + 23\right)\cdot 31 + \left(15 a + 14\right)\cdot 31^{2} + \left(a + 30\right)\cdot 31^{3} + \left(15 a + 2\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 27 a + 17 + \left(8 a + 21\right)\cdot 31 + \left(26 a + 24\right)\cdot 31^{2} + \left(27 a + 7\right)\cdot 31^{3} + \left(23 a + 14\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 4 a + 9 + \left(22 a + 12\right)\cdot 31 + \left(4 a + 6\right)\cdot 31^{2} + \left(3 a + 6\right)\cdot 31^{3} + \left(7 a + 3\right)\cdot 31^{4} +O(31^{5})\)
|
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.