Basic invariants
| Dimension: | $3$ |
| Group: | $S_4\times C_2$ |
| Conductor: | \(10245312\)\(\medspace = 2^{6} \cdot 3^{3} \cdot 7^{2} \cdot 11^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.10245312.4 |
| 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 4.2.133056.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} + 2x^{4} + 15x^{2} + 3 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 61 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 61 }$:
\( x^{2} + 60x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 57 a + 2 + \left(7 a + 55\right)\cdot 61 + \left(15 a + 26\right)\cdot 61^{2} + \left(3 a + 36\right)\cdot 61^{3} + \left(36 a + 44\right)\cdot 61^{4} + \left(26 a + 4\right)\cdot 61^{5} + \left(52 a + 48\right)\cdot 61^{6} + \left(57 a + 27\right)\cdot 61^{7} + \left(29 a + 44\right)\cdot 61^{8} +O(61^{9})\)
|
| $r_{ 2 }$ | $=$ |
\( 18 + 43\cdot 61 + 27\cdot 61^{2} + 13\cdot 61^{3} + 24\cdot 61^{4} + 30\cdot 61^{5} + 55\cdot 61^{6} + 34\cdot 61^{7} + 41\cdot 61^{8} +O(61^{9})\)
|
| $r_{ 3 }$ | $=$ |
\( 39 a + 11 + \left(22 a + 8\right)\cdot 61 + \left(49 a + 17\right)\cdot 61^{2} + \left(20 a + 14\right)\cdot 61^{3} + \left(12 a + 4\right)\cdot 61^{4} + \left(5 a + 34\right)\cdot 61^{5} + \left(39 a + 13\right)\cdot 61^{6} + \left(23 a + 38\right)\cdot 61^{7} + \left(60 a + 42\right)\cdot 61^{8} +O(61^{9})\)
|
| $r_{ 4 }$ | $=$ |
\( 4 a + 59 + \left(53 a + 5\right)\cdot 61 + \left(45 a + 34\right)\cdot 61^{2} + \left(57 a + 24\right)\cdot 61^{3} + \left(24 a + 16\right)\cdot 61^{4} + \left(34 a + 56\right)\cdot 61^{5} + \left(8 a + 12\right)\cdot 61^{6} + \left(3 a + 33\right)\cdot 61^{7} + \left(31 a + 16\right)\cdot 61^{8} +O(61^{9})\)
|
| $r_{ 5 }$ | $=$ |
\( 43 + 17\cdot 61 + 33\cdot 61^{2} + 47\cdot 61^{3} + 36\cdot 61^{4} + 30\cdot 61^{5} + 5\cdot 61^{6} + 26\cdot 61^{7} + 19\cdot 61^{8} +O(61^{9})\)
|
| $r_{ 6 }$ | $=$ |
\( 22 a + 50 + \left(38 a + 52\right)\cdot 61 + \left(11 a + 43\right)\cdot 61^{2} + \left(40 a + 46\right)\cdot 61^{3} + \left(48 a + 56\right)\cdot 61^{4} + \left(55 a + 26\right)\cdot 61^{5} + \left(21 a + 47\right)\cdot 61^{6} + \left(37 a + 22\right)\cdot 61^{7} + 18\cdot 61^{8} +O(61^{9})\)
|
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 |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-3$ |
| $3$ | $2$ | $(2,5)$ | $1$ |
| $3$ | $2$ | $(2,5)(3,6)$ | $-1$ |
| $6$ | $2$ | $(1,3)(4,6)$ | $-1$ |
| $6$ | $2$ | $(1,3)(2,5)(4,6)$ | $1$ |
| $8$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $6$ | $4$ | $(2,6,5,3)$ | $-1$ |
| $6$ | $4$ | $(1,4)(2,6,5,3)$ | $1$ |
| $8$ | $6$ | $(1,2,6,4,5,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.