Basic invariants
| Dimension: | $3$ |
| Group: | $S_4$ |
| Conductor: | \(65108761\)\(\medspace = 8069^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 4.4.8069.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_4$ |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.4.8069.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} - x^{3} - 5x^{2} + 5x + 1 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 6 + 2\cdot 7^{2} + 4\cdot 7^{4} +O(7^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 4 + 6\cdot 7 + 3\cdot 7^{2} + 3\cdot 7^{3} + 7^{4} +O(7^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 5 a + \left(a + 5\right)\cdot 7 + a\cdot 7^{2} + \left(6 a + 6\right)\cdot 7^{3} + \left(3 a + 1\right)\cdot 7^{4} +O(7^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 2 a + 5 + \left(5 a + 1\right)\cdot 7 + 5 a\cdot 7^{2} + 4\cdot 7^{3} + \left(3 a + 6\right)\cdot 7^{4} +O(7^{5})\)
|
Generators of the action on the roots $r_1, \ldots, r_{ 4 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 4 }$ | Character value |
| $1$ | $1$ | $()$ | $3$ |
| $3$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $6$ | $2$ | $(1,2)$ | $-1$ |
| $8$ | $3$ | $(1,2,3)$ | $0$ |
| $6$ | $4$ | $(1,2,3,4)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.