Basic invariants
| Dimension: | $3$ |
| Group: | $S_4\times C_2$ |
| Conductor: | \(6429888\)\(\medspace = 2^{6} \cdot 3^{3} \cdot 61^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.6429888.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_4\times C_2$ |
| Parity: | odd |
| Determinant: | 1.3.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.2.13176.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 3x^{5} + 2x^{4} + x^{3} + 15x^{2} - 16x + 12 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 67 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 67 }$:
\( x^{2} + 63x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 28 a + 45 + \left(55 a + 3\right)\cdot 67 + \left(12 a + 2\right)\cdot 67^{2} + \left(65 a + 10\right)\cdot 67^{3} + \left(30 a + 4\right)\cdot 67^{4} + \left(30 a + 55\right)\cdot 67^{5} + \left(9 a + 29\right)\cdot 67^{6} + 2 a\cdot 67^{7} +O(67^{8})\)
|
| $r_{ 2 }$ | $=$ |
\( 10 a + 14 + \left(48 a + 9\right)\cdot 67 + \left(52 a + 19\right)\cdot 67^{2} + \left(45 a + 35\right)\cdot 67^{3} + \left(18 a + 52\right)\cdot 67^{4} + \left(31 a + 13\right)\cdot 67^{5} + \left(41 a + 33\right)\cdot 67^{6} + \left(7 a + 5\right)\cdot 67^{7} +O(67^{8})\)
|
| $r_{ 3 }$ | $=$ |
\( 39 a + 23 + \left(11 a + 63\right)\cdot 67 + \left(54 a + 64\right)\cdot 67^{2} + \left(a + 56\right)\cdot 67^{3} + \left(36 a + 62\right)\cdot 67^{4} + \left(36 a + 11\right)\cdot 67^{5} + \left(57 a + 37\right)\cdot 67^{6} + \left(64 a + 66\right)\cdot 67^{7} +O(67^{8})\)
|
| $r_{ 4 }$ | $=$ |
\( 57 a + 54 + \left(18 a + 57\right)\cdot 67 + \left(14 a + 47\right)\cdot 67^{2} + \left(21 a + 31\right)\cdot 67^{3} + \left(48 a + 14\right)\cdot 67^{4} + \left(35 a + 53\right)\cdot 67^{5} + \left(25 a + 33\right)\cdot 67^{6} + \left(59 a + 61\right)\cdot 67^{7} +O(67^{8})\)
|
| $r_{ 5 }$ | $=$ |
\( 15 + 32\cdot 67 + 34\cdot 67^{2} + 10\cdot 67^{3} + 40\cdot 67^{4} + 50\cdot 67^{5} + 43\cdot 67^{6} + 48\cdot 67^{7} +O(67^{8})\)
|
| $r_{ 6 }$ | $=$ |
\( 53 + 34\cdot 67 + 32\cdot 67^{2} + 56\cdot 67^{3} + 26\cdot 67^{4} + 16\cdot 67^{5} + 23\cdot 67^{6} + 18\cdot 67^{7} +O(67^{8})\)
|
Generators of the action on the roots $r_1, \ldots, r_{ 6 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 6 }$ | Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,3)(2,4)(5,6)$ | $-3$ |
| $3$ | $2$ | $(1,3)$ | $1$ |
| $3$ | $2$ | $(1,3)(2,4)$ | $-1$ |
| $6$ | $2$ | $(2,5)(4,6)$ | $-1$ |
| $6$ | $2$ | $(1,3)(2,5)(4,6)$ | $1$ |
| $8$ | $3$ | $(1,2,5)(3,4,6)$ | $0$ |
| $6$ | $4$ | $(1,4,3,2)$ | $-1$ |
| $6$ | $4$ | $(1,6,3,5)(2,4)$ | $1$ |
| $8$ | $6$ | $(1,4,6,3,2,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.