Basic invariants
| Dimension: | $4$ |
| Group: | $S_5$ |
| Conductor: | \(3857\)\(\medspace = 7 \cdot 19 \cdot 29 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.1.3857.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_5$ |
| Parity: | even |
| Determinant: | 1.3857.2t1.a.a |
| Projective image: | $S_5$ |
| Projective stem field: | Galois closure of 5.1.3857.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - x^{4} + x^{3} - x^{2} + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 61 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 61 }$:
\( x^{2} + 60x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 10 a + 57 + \left(37 a + 10\right)\cdot 61 + \left(9 a + 57\right)\cdot 61^{2} + 33\cdot 61^{3} + \left(4 a + 4\right)\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 51 a + 6 + \left(23 a + 38\right)\cdot 61 + \left(51 a + 29\right)\cdot 61^{2} + \left(60 a + 24\right)\cdot 61^{3} + \left(56 a + 8\right)\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 31 a + 11 + \left(53 a + 28\right)\cdot 61 + \left(30 a + 43\right)\cdot 61^{2} + \left(8 a + 37\right)\cdot 61^{3} + \left(21 a + 3\right)\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 7 + 55\cdot 61 + 31\cdot 61^{2} + 10\cdot 61^{3} + 28\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 30 a + 42 + \left(7 a + 50\right)\cdot 61 + \left(30 a + 20\right)\cdot 61^{2} + \left(52 a + 15\right)\cdot 61^{3} + \left(39 a + 16\right)\cdot 61^{4} +O(61^{5})\)
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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 | Complex conjugation |
| $1$ | $1$ | $()$ | $4$ | |
| $10$ | $2$ | $(1,2)$ | $2$ | |
| $15$ | $2$ | $(1,2)(3,4)$ | $0$ | ✓ |
| $20$ | $3$ | $(1,2,3)$ | $1$ | |
| $30$ | $4$ | $(1,2,3,4)$ | $0$ | |
| $24$ | $5$ | $(1,2,3,4,5)$ | $-1$ | |
| $20$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |