Basic invariants
| Dimension: | $2$ |
| Group: | $S_3\times C_3$ |
| Conductor: | \(3969\)\(\medspace = 3^{4} \cdot 7^{2} \) |
| Artin stem field: | Galois closure of 6.0.110270727.3 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $S_3\times C_3$ |
| Parity: | odd |
| Determinant: | 1.63.6t1.d.a |
| Projective image: | $S_3$ |
| Projective stem field: | Galois closure of 3.1.567.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 7x^{3} + 63x^{2} + 21x + 14 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 101 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 101 }$:
\( x^{2} + 97x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 16 a + 82 + \left(51 a + 55\right)\cdot 101 + \left(60 a + 3\right)\cdot 101^{2} + \left(62 a + 53\right)\cdot 101^{3} + \left(75 a + 13\right)\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 90 a + 97 + \left(47 a + 48\right)\cdot 101 + \left(43 a + 30\right)\cdot 101^{2} + \left(84 a + 30\right)\cdot 101^{3} + \left(24 a + 26\right)\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 5 a + 3 + \left(99 a + 9\right)\cdot 101 + \left(2 a + 53\right)\cdot 101^{2} + \left(46 a + 38\right)\cdot 101^{3} + \left(100 a + 8\right)\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 85 a + 45 + \left(49 a + 42\right)\cdot 101 + \left(40 a + 93\right)\cdot 101^{2} + \left(38 a + 40\right)\cdot 101^{3} + \left(25 a + 51\right)\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 11 a + 53 + \left(53 a + 49\right)\cdot 101 + \left(57 a + 55\right)\cdot 101^{2} + \left(16 a + 21\right)\cdot 101^{3} + \left(76 a + 41\right)\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 96 a + 23 + \left(a + 97\right)\cdot 101 + \left(98 a + 66\right)\cdot 101^{2} + \left(54 a + 17\right)\cdot 101^{3} + 61\cdot 101^{4} +O(101^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $3$ | $2$ | $(1,3)(2,4)(5,6)$ | $0$ |
| $1$ | $3$ | $(1,2,6)(3,4,5)$ | $2 \zeta_{3}$ |
| $1$ | $3$ | $(1,6,2)(3,5,4)$ | $-2 \zeta_{3} - 2$ |
| $2$ | $3$ | $(1,2,6)(3,5,4)$ | $-1$ |
| $2$ | $3$ | $(1,2,6)$ | $\zeta_{3} + 1$ |
| $2$ | $3$ | $(1,6,2)$ | $-\zeta_{3}$ |
| $3$ | $6$ | $(1,3,2,4,6,5)$ | $0$ |
| $3$ | $6$ | $(1,5,6,4,2,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.