Basic invariants
| Dimension: | $4$ |
| Group: | $S_5$ |
| Conductor: | \(4477\)\(\medspace = 11^{2} \cdot 37 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.1.4477.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_5$ |
| Parity: | even |
| Determinant: | 1.37.2t1.a.a |
| Projective image: | $S_5$ |
| Projective stem field: | Galois closure of 5.1.4477.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} + x^{3} + x - 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 101 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 101 }$:
\( x^{2} + 97x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 90 a + 50 + \left(29 a + 88\right)\cdot 101 + \left(89 a + 37\right)\cdot 101^{2} + \left(83 a + 87\right)\cdot 101^{3} + \left(36 a + 82\right)\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 11 a + 6 + \left(71 a + 17\right)\cdot 101 + \left(11 a + 62\right)\cdot 101^{2} + \left(17 a + 30\right)\cdot 101^{3} + \left(64 a + 45\right)\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 48 + 31\cdot 101 + 100\cdot 101^{2} + 32\cdot 101^{3} + 5\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 72 a + 6 + \left(42 a + 84\right)\cdot 101 + \left(46 a + 80\right)\cdot 101^{2} + \left(83 a + 83\right)\cdot 101^{3} + \left(76 a + 73\right)\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 29 a + 92 + \left(58 a + 81\right)\cdot 101 + \left(54 a + 21\right)\cdot 101^{2} + \left(17 a + 68\right)\cdot 101^{3} + \left(24 a + 95\right)\cdot 101^{4} +O(101^{5})\)
|
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 | Complex conjugation |
| $1$ | $1$ | $()$ | $4$ | |
| $10$ | $2$ | $(1,2)$ | $2$ | |
| $15$ | $2$ | $(1,2)(3,4)$ | $0$ | ✓ |
| $20$ | $3$ | $(1,2,3)$ | $1$ | |
| $30$ | $4$ | $(1,2,3,4)$ | $0$ | |
| $24$ | $5$ | $(1,2,3,4,5)$ | $-1$ | |
| $20$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |