Basic invariants
| Dimension: | $3$ |
| Group: | $S_4$ |
| Conductor: | \(17926756\)\(\medspace = 2^{2} \cdot 29^{2} \cdot 73^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 4.4.8468.1 |
| 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 4.4.8468.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} - x^{3} - 5x^{2} + 3x + 4 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$:
\( x^{2} + 29x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 26 a + 1 + \left(30 a + 1\right)\cdot 31 + \left(5 a + 26\right)\cdot 31^{2} + \left(8 a + 15\right)\cdot 31^{3} + \left(18 a + 26\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 8 a + 12 + \left(25 a + 6\right)\cdot 31 + \left(19 a + 7\right)\cdot 31^{2} + \left(24 a + 26\right)\cdot 31^{3} + \left(5 a + 27\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 5 a + 22 + 5\cdot 31 + \left(25 a + 7\right)\cdot 31^{2} + \left(22 a + 26\right)\cdot 31^{3} + \left(12 a + 23\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 23 a + 28 + \left(5 a + 17\right)\cdot 31 + \left(11 a + 21\right)\cdot 31^{2} + \left(6 a + 24\right)\cdot 31^{3} + \left(25 a + 14\right)\cdot 31^{4} +O(31^{5})\)
|
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.