Basic invariants
| Dimension: | $4$ |
| Group: | $S_5$ |
| Conductor: | \(58818484369\)\(\medspace = 3889^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.1.3889.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_5$ |
| Parity: | even |
| Determinant: | 1.3889.2t1.a.a |
| Projective image: | $S_5$ |
| Projective stem field: | Galois closure of 5.1.3889.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - x^{4} + x^{3} + x - 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$:
\( x^{2} + 29x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 15 a + 22 + \left(17 a + 28\right)\cdot 31 + \left(24 a + 7\right)\cdot 31^{2} + \left(29 a + 12\right)\cdot 31^{3} + \left(27 a + 5\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 27 a + 19 + \left(20 a + 12\right)\cdot 31 + \left(2 a + 25\right)\cdot 31^{2} + \left(20 a + 19\right)\cdot 31^{3} + \left(9 a + 25\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 16 a + 21 + \left(13 a + 17\right)\cdot 31 + \left(6 a + 8\right)\cdot 31^{2} + \left(a + 16\right)\cdot 31^{3} + 3 a\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 4 a + 11 + \left(10 a + 27\right)\cdot 31 + \left(28 a + 9\right)\cdot 31^{2} + \left(10 a + 26\right)\cdot 31^{3} + \left(21 a + 24\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 21 + 6\cdot 31 + 10\cdot 31^{2} + 18\cdot 31^{3} + 5\cdot 31^{4} +O(31^{5})\)
|
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.