Basic invariants
| Dimension: | $3$ |
| Group: | $S_4$ |
| Conductor: | \(1444\)\(\medspace = 2^{2} \cdot 19^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 4.2.27436.1 |
| 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 4.2.27436.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} - x^{3} - 2x^{2} - 6x - 2 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$:
\( x^{2} + 12x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 9 a + 6 + \left(3 a + 12\right)\cdot 13 + \left(11 a + 5\right)\cdot 13^{2} + \left(4 a + 8\right)\cdot 13^{3} + \left(5 a + 10\right)\cdot 13^{4} +O(13^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 8 + 13 + 12\cdot 13^{2} + 3\cdot 13^{3} + 13^{4} +O(13^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 11 + 4\cdot 13 + 7\cdot 13^{2} + 11\cdot 13^{3} + 2\cdot 13^{4} +O(13^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 4 a + 2 + \left(9 a + 7\right)\cdot 13 + a\cdot 13^{2} + \left(8 a + 2\right)\cdot 13^{3} + \left(7 a + 11\right)\cdot 13^{4} +O(13^{5})\)
|
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 | Complex conjugation |
| $1$ | $1$ | $()$ | $3$ | |
| $3$ | $2$ | $(1,2)(3,4)$ | $-1$ | |
| $6$ | $2$ | $(1,2)$ | $-1$ | ✓ |
| $8$ | $3$ | $(1,2,3)$ | $0$ | |
| $6$ | $4$ | $(1,2,3,4)$ | $1$ |