Basic invariants
| Dimension: | $6$ |
| Group: | $S_5$ |
| Conductor: | \(6884147200000000\)\(\medspace = 2^{20} \cdot 5^{8} \cdot 7^{5} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.1.137200000000.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 20T30 |
| Parity: | even |
| Determinant: | 1.28.2t1.a.a |
| Projective image: | $S_5$ |
| Projective stem field: | Galois closure of 5.1.137200000000.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - 50x^{3} - 1200x^{2} - 6625x - 20200 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 14 a + 12 + \left(15 a + 17\right)\cdot 19 + \left(3 a + 9\right)\cdot 19^{2} + \left(18 a + 9\right)\cdot 19^{3} + \left(11 a + 16\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 14 a + \left(12 a + 12\right)\cdot 19 + \left(5 a + 6\right)\cdot 19^{2} + \left(12 a + 12\right)\cdot 19^{3} + \left(17 a + 17\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 5 a + 14 + \left(6 a + 10\right)\cdot 19 + \left(13 a + 18\right)\cdot 19^{2} + \left(6 a + 18\right)\cdot 19^{3} + \left(a + 3\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 5 a + 7 + 3 a\cdot 19 + \left(15 a + 17\right)\cdot 19^{2} + 4\cdot 19^{3} + \left(7 a + 10\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 5 + 16\cdot 19 + 4\cdot 19^{2} + 11\cdot 19^{3} + 8\cdot 19^{4} +O(19^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $6$ |
| $10$ | $2$ | $(1,2)$ | $0$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $-2$ |
| $20$ | $3$ | $(1,2,3)$ | $0$ |
| $30$ | $4$ | $(1,2,3,4)$ | $0$ |
| $24$ | $5$ | $(1,2,3,4,5)$ | $1$ |
| $20$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.