Basic invariants
| Dimension: | $5$ |
| Group: | $S_5$ |
| Conductor: | \(5075375625\)\(\medspace = 3^{3} \cdot 5^{4} \cdot 67^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.1.5025.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $\PGL(2,5)$ |
| Parity: | even |
| Determinant: | 1.201.2t1.a.a |
| Projective image: | $S_5$ |
| Projective stem field: | Galois closure of 5.1.5025.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - x^{4} - x^{2} + x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 71 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 71 }$:
\( x^{2} + 69x + 7 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 42 a + 34 + \left(31 a + 23\right)\cdot 71 + \left(13 a + 44\right)\cdot 71^{2} + \left(68 a + 53\right)\cdot 71^{3} + \left(39 a + 12\right)\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 32 + 12\cdot 71 + 6\cdot 71^{2} + 68\cdot 71^{3} + 58\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 29 a + 47 + \left(39 a + 44\right)\cdot 71 + \left(57 a + 39\right)\cdot 71^{2} + \left(2 a + 34\right)\cdot 71^{3} + \left(31 a + 24\right)\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 57 + 38\cdot 71^{2} + 60\cdot 71^{3} + 27\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 44 + 60\cdot 71 + 13\cdot 71^{2} + 67\cdot 71^{3} + 17\cdot 71^{4} +O(71^{5})\)
|
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.