Basic invariants
| Dimension: | $5$ |
| Group: | $S_5$ |
| Conductor: | \(253414561\)\(\medspace = 15919^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.3.15919.1 |
| 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 5.3.15919.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - 2x^{3} - 3x^{2} + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 127 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 127 }$:
\( x^{2} + 126x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 15 a + 12 + \left(19 a + 37\right)\cdot 127 + \left(105 a + 23\right)\cdot 127^{2} + \left(105 a + 40\right)\cdot 127^{3} + \left(17 a + 102\right)\cdot 127^{4} +O(127^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 14 + 27\cdot 127 + 65\cdot 127^{2} + 20\cdot 127^{3} + 13\cdot 127^{4} +O(127^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 112 a + 27 + \left(107 a + 41\right)\cdot 127 + \left(21 a + 109\right)\cdot 127^{2} + \left(21 a + 40\right)\cdot 127^{3} + \left(109 a + 14\right)\cdot 127^{4} +O(127^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 18 + 22\cdot 127 + 12\cdot 127^{2} + 9\cdot 127^{3} + 91\cdot 127^{4} +O(127^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 56 + 126\cdot 127 + 43\cdot 127^{2} + 16\cdot 127^{3} + 33\cdot 127^{4} +O(127^{5})\)
|
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.