Basic invariants
| Dimension: | $4$ |
| Group: | $S_5$ |
| Conductor: | \(6581\) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.1.6581.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_5$ |
| Parity: | even |
| Determinant: | 1.6581.2t1.a.a |
| Projective image: | $S_5$ |
| Projective stem field: | Galois closure of 5.1.6581.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - 2x^{2} - 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 7 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 7 }$:
\( x^{2} + 6x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 3 + 3\cdot 7 + 5\cdot 7^{2} + 3\cdot 7^{3} + 3\cdot 7^{4} +O(7^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 3 a + 5 + \left(6 a + 4\right)\cdot 7 + \left(6 a + 5\right)\cdot 7^{2} + \left(3 a + 2\right)\cdot 7^{3} + 7^{4} +O(7^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 3 a + 1 + \left(3 a + 2\right)\cdot 7 + 6 a\cdot 7^{2} + 3\cdot 7^{3} + 2\cdot 7^{4} +O(7^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 4 a + 1 + 7 + 6\cdot 7^{2} + \left(3 a + 6\right)\cdot 7^{3} + \left(6 a + 4\right)\cdot 7^{4} +O(7^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 4 a + 4 + \left(3 a + 2\right)\cdot 7 + 3\cdot 7^{2} + \left(6 a + 4\right)\cdot 7^{3} + \left(6 a + 1\right)\cdot 7^{4} +O(7^{5})\)
|
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 | Complex conjugation |
| $1$ | $1$ | $()$ | $4$ | |
| $10$ | $2$ | $(1,2)$ | $2$ | |
| $15$ | $2$ | $(1,2)(3,4)$ | $0$ | ✓ |
| $20$ | $3$ | $(1,2,3)$ | $1$ | |
| $30$ | $4$ | $(1,2,3,4)$ | $0$ | |
| $24$ | $5$ | $(1,2,3,4,5)$ | $-1$ | |
| $20$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |