Basic invariants
| Dimension: | $5$ |
| Group: | $S_5$ |
| Conductor: | \(31494544\)\(\medspace = 2^{4} \cdot 23^{2} \cdot 61^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.3.22448.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_5$ |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $S_5$ |
| Projective stem field: | Galois closure of 5.3.22448.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - x^{4} - 2x^{3} + 2x^{2} - x - 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$:
\( x^{2} + 24x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 4 + 12\cdot 29 + 29^{2} + 17\cdot 29^{3} + 10\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 4 a + 6 + \left(27 a + 20\right)\cdot 29 + \left(16 a + 13\right)\cdot 29^{2} + \left(26 a + 28\right)\cdot 29^{3} + \left(28 a + 9\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 13 + 19\cdot 29 + 2\cdot 29^{2} + 7\cdot 29^{3} +O(29^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 10 + 28\cdot 29 + 26\cdot 29^{2} + 5\cdot 29^{3} + 25\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 25 a + 26 + \left(a + 6\right)\cdot 29 + \left(12 a + 13\right)\cdot 29^{2} + \left(2 a + 28\right)\cdot 29^{3} + 11\cdot 29^{4} +O(29^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $5$ |
| $10$ | $2$ | $(1,2)$ | $1$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $20$ | $3$ | $(1,2,3)$ | $-1$ |
| $30$ | $4$ | $(1,2,3,4)$ | $-1$ |
| $24$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $20$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.