Basic invariants
| Dimension: | $4$ |
| Group: | $S_5$ |
| Conductor: | \(2869\)\(\medspace = 19 \cdot 151 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.1.2869.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_5$ |
| Parity: | even |
| Determinant: | 1.2869.2t1.a.a |
| Projective image: | $S_5$ |
| Projective stem field: | Galois closure of 5.1.2869.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - x - 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 67 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 67 }$:
\( x^{2} + 63x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 37 a + 11 + \left(52 a + 51\right)\cdot 67 + \left(a + 63\right)\cdot 67^{2} + \left(31 a + 51\right)\cdot 67^{3} + 43\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 34 a + 41 + \left(44 a + 28\right)\cdot 67 + \left(12 a + 3\right)\cdot 67^{2} + \left(18 a + 16\right)\cdot 67^{3} + \left(36 a + 60\right)\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 14 + 59\cdot 67 + 38\cdot 67^{2} + 16\cdot 67^{3} + 29\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 30 a + 25 + \left(14 a + 23\right)\cdot 67 + \left(65 a + 18\right)\cdot 67^{2} + \left(35 a + 40\right)\cdot 67^{3} + \left(66 a + 14\right)\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 33 a + 43 + \left(22 a + 38\right)\cdot 67 + \left(54 a + 9\right)\cdot 67^{2} + \left(48 a + 9\right)\cdot 67^{3} + \left(30 a + 53\right)\cdot 67^{4} +O(67^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $4$ | |
| $10$ | $2$ | $(1,2)$ | $2$ | |
| $15$ | $2$ | $(1,2)(3,4)$ | $0$ | ✓ |
| $20$ | $3$ | $(1,2,3)$ | $1$ | |
| $30$ | $4$ | $(1,2,3,4)$ | $0$ | |
| $24$ | $5$ | $(1,2,3,4,5)$ | $-1$ | |
| $20$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |