Basic invariants
| Dimension: | $5$ |
| Group: | $S_5$ |
| Conductor: | \(3131073936\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 4663^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.5.223824.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_5$ |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $S_5$ |
| Projective stem field: | Galois closure of 5.5.223824.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - 2x^{4} - 4x^{3} + 6x^{2} + 2x - 2 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 23 + 48\cdot 67^{2} + 7\cdot 67^{3} + 23\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 25 + 44\cdot 67 + 27\cdot 67^{2} + 14\cdot 67^{3} + 65\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 20 a + 17 + \left(5 a + 18\right)\cdot 67 + \left(51 a + 63\right)\cdot 67^{2} + \left(18 a + 65\right)\cdot 67^{3} + \left(25 a + 43\right)\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 47 a + 30 + \left(61 a + 19\right)\cdot 67 + \left(15 a + 61\right)\cdot 67^{2} + \left(48 a + 22\right)\cdot 67^{3} + \left(41 a + 59\right)\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 41 + 51\cdot 67 + 23\cdot 67^{3} + 9\cdot 67^{4} +O(67^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $5$ |
| $10$ | $2$ | $(1,2)$ | $1$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $20$ | $3$ | $(1,2,3)$ | $-1$ |
| $30$ | $4$ | $(1,2,3,4)$ | $-1$ |
| $24$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $20$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.