Basic invariants
| Dimension: | $4$ |
| Group: | $S_5$ |
| Conductor: | \(147109\)\(\medspace = 157 \cdot 937 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.5.147109.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_5$ |
| Parity: | even |
| Determinant: | 1.147109.2t1.a.a |
| Projective image: | $S_5$ |
| Projective stem field: | Galois closure of 5.5.147109.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - 2x^{4} - 4x^{3} + 5x^{2} + 3x - 2 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 59 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 59 }$:
\( x^{2} + 58x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 52 a + 5 + \left(44 a + 18\right)\cdot 59 + 51\cdot 59^{2} + \left(35 a + 19\right)\cdot 59^{3} + \left(42 a + 1\right)\cdot 59^{4} +O(59^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 18 a + 27 + \left(48 a + 32\right)\cdot 59 + \left(47 a + 23\right)\cdot 59^{2} + \left(48 a + 38\right)\cdot 59^{3} + \left(53 a + 12\right)\cdot 59^{4} +O(59^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 7 a + 57 + \left(14 a + 10\right)\cdot 59 + \left(58 a + 7\right)\cdot 59^{2} + \left(23 a + 54\right)\cdot 59^{3} + \left(16 a + 8\right)\cdot 59^{4} +O(59^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 41 a + 45 + \left(10 a + 3\right)\cdot 59 + \left(11 a + 23\right)\cdot 59^{2} + \left(10 a + 39\right)\cdot 59^{3} + \left(5 a + 17\right)\cdot 59^{4} +O(59^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 45 + 52\cdot 59 + 12\cdot 59^{2} + 25\cdot 59^{3} + 18\cdot 59^{4} +O(59^{5})\)
|
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.