Basic invariants
| Dimension: | $5$ |
| Group: | $S_5$ |
| Conductor: | \(152236964367\)\(\medspace = 3^{2} \cdot 17^{3} \cdot 151^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.3.23103.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $\PGL(2,5)$ |
| Parity: | odd |
| Determinant: | 1.2567.2t1.a.a |
| Projective image: | $S_5$ |
| Projective stem field: | Galois closure of 5.3.23103.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - x^{4} + x^{2} - 3x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 103 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 103 }$:
\( x^{2} + 102x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 22 + 96\cdot 103 + 36\cdot 103^{2} + 93\cdot 103^{3} + 2\cdot 103^{4} +O(103^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 39 a + 63 + \left(61 a + 89\right)\cdot 103 + \left(49 a + 33\right)\cdot 103^{2} + \left(28 a + 5\right)\cdot 103^{3} + \left(84 a + 15\right)\cdot 103^{4} +O(103^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 55 a + 34 + \left(16 a + 76\right)\cdot 103 + \left(43 a + 94\right)\cdot 103^{2} + \left(75 a + 96\right)\cdot 103^{3} + \left(a + 43\right)\cdot 103^{4} +O(103^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 48 a + 89 + \left(86 a + 37\right)\cdot 103 + \left(59 a + 18\right)\cdot 103^{2} + \left(27 a + 26\right)\cdot 103^{3} + \left(101 a + 73\right)\cdot 103^{4} +O(103^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 64 a + 102 + \left(41 a + 8\right)\cdot 103 + \left(53 a + 22\right)\cdot 103^{2} + \left(74 a + 87\right)\cdot 103^{3} + \left(18 a + 70\right)\cdot 103^{4} +O(103^{5})\)
|
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.