Basic invariants
| Dimension: | $5$ |
| Group: | $S_5$ |
| Conductor: | \(14625792\)\(\medspace = 2^{10} \cdot 3^{3} \cdot 23^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.3.3656448.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $\PGL(2,5)$ |
| Parity: | odd |
| Determinant: | 1.3.2t1.a.a |
| Projective image: | $S_5$ |
| Projective stem field: | Galois closure of 5.3.3656448.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - 10x^{3} - 4x^{2} + 27x + 16 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 3 a + 23 + \left(11 a + 27\right)\cdot 29 + \left(21 a + 27\right)\cdot 29^{2} + \left(10 a + 3\right)\cdot 29^{3} + \left(5 a + 11\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 19 + 9\cdot 29 + 7\cdot 29^{2} + 12\cdot 29^{3} + 10\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 14 + 10\cdot 29 + 23\cdot 29^{2} + 14\cdot 29^{3} + 10\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 26 a + 9 + \left(17 a + 22\right)\cdot 29 + \left(7 a + 7\right)\cdot 29^{2} + \left(18 a + 7\right)\cdot 29^{3} + \left(23 a + 27\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 22 + 16\cdot 29 + 20\cdot 29^{2} + 19\cdot 29^{3} + 27\cdot 29^{4} +O(29^{5})\)
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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 |
| $1$ | $1$ | $()$ | $5$ |
| $10$ | $2$ | $(1,2)$ | $-1$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $20$ | $3$ | $(1,2,3)$ | $-1$ |
| $30$ | $4$ | $(1,2,3,4)$ | $1$ |
| $24$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $20$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.