Basic invariants
| Dimension: | $4$ |
| Group: | $S_5$ |
| Conductor: | \(3315280740031\)\(\medspace = 13^{3} \cdot 31^{3} \cdot 37^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.3.14911.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_5$ |
| Parity: | odd |
| Determinant: | 1.14911.2t1.a.a |
| Projective image: | $S_5$ |
| Projective stem field: | Galois closure of 5.3.14911.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - x^{4} - 2x^{3} - x^{2} + 3x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$:
\( x^{2} + 18x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 7 a + 6 + \left(12 a + 15\right)\cdot 19 + \left(6 a + 10\right)\cdot 19^{2} + \left(14 a + 12\right)\cdot 19^{3} + 8 a\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 3 a + 4 + \left(17 a + 14\right)\cdot 19 + \left(9 a + 10\right)\cdot 19^{2} + \left(16 a + 12\right)\cdot 19^{3} + \left(3 a + 1\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 9 + 16\cdot 19 + 7\cdot 19^{2} + 11\cdot 19^{3} + 13\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 12 a + 13 + \left(6 a + 1\right)\cdot 19 + \left(12 a + 5\right)\cdot 19^{2} + \left(4 a + 1\right)\cdot 19^{3} + \left(10 a + 14\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 16 a + 7 + \left(a + 9\right)\cdot 19 + \left(9 a + 3\right)\cdot 19^{2} + 2 a\cdot 19^{3} + \left(15 a + 8\right)\cdot 19^{4} +O(19^{5})\)
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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 |
| $1$ | $1$ | $()$ | $4$ |
| $10$ | $2$ | $(1,2)$ | $-2$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $20$ | $3$ | $(1,2,3)$ | $1$ |
| $30$ | $4$ | $(1,2,3,4)$ | $0$ |
| $24$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
| $20$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.