Basic invariants
| Dimension: | $2$ |
| Group: | $S_3\times C_3$ |
| Conductor: | \(140\)\(\medspace = 2^{2} \cdot 5 \cdot 7 \) |
| Artin stem field: | Galois closure of 6.0.392000.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $S_3\times C_3$ |
| Parity: | odd |
| Determinant: | 1.140.6t1.b.a |
| Projective image: | $S_3$ |
| Projective stem field: | Galois closure of 3.1.980.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} + 3x^{4} + 4x^{2} + 2x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$:
\( x^{2} + 12x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 12 a + 6 + \left(7 a + 4\right)\cdot 13 + \left(9 a + 1\right)\cdot 13^{2} + \left(9 a + 8\right)\cdot 13^{3} + \left(7 a + 3\right)\cdot 13^{4} + \left(11 a + 10\right)\cdot 13^{5} + \left(6 a + 2\right)\cdot 13^{6} +O(13^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 7 a + 11 + \left(10 a + 8\right)\cdot 13 + \left(3 a + 8\right)\cdot 13^{2} + 9\cdot 13^{3} + \left(3 a + 7\right)\cdot 13^{4} + \left(2 a + 7\right)\cdot 13^{5} + \left(12 a + 12\right)\cdot 13^{6} +O(13^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( a + 5 + 5 a\cdot 13 + \left(3 a + 3\right)\cdot 13^{2} + \left(3 a + 8\right)\cdot 13^{3} + \left(5 a + 1\right)\cdot 13^{4} + \left(a + 1\right)\cdot 13^{5} + \left(6 a + 11\right)\cdot 13^{6} +O(13^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 6 a + 5 + \left(2 a + 12\right)\cdot 13 + \left(9 a + 1\right)\cdot 13^{2} + \left(12 a + 6\right)\cdot 13^{3} + \left(9 a + 10\right)\cdot 13^{4} + \left(10 a + 6\right)\cdot 13^{5} + 9\cdot 13^{6} +O(13^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( 8 a + 3 + \left(2 a + 9\right)\cdot 13 + \left(7 a + 9\right)\cdot 13^{2} + \left(3 a + 11\right)\cdot 13^{3} + \left(8 a + 11\right)\cdot 13^{4} + \left(3 a + 8\right)\cdot 13^{5} + 5 a\cdot 13^{6} +O(13^{7})\)
|
| $r_{ 6 }$ | $=$ |
\( 5 a + 11 + \left(10 a + 3\right)\cdot 13 + \left(5 a + 1\right)\cdot 13^{2} + \left(9 a + 8\right)\cdot 13^{3} + \left(4 a + 3\right)\cdot 13^{4} + \left(9 a + 4\right)\cdot 13^{5} + \left(7 a + 2\right)\cdot 13^{6} +O(13^{7})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $3$ | $2$ | $(1,3)(2,4)(5,6)$ | $0$ |
| $1$ | $3$ | $(1,4,5)(2,6,3)$ | $2 \zeta_{3}$ |
| $1$ | $3$ | $(1,5,4)(2,3,6)$ | $-2 \zeta_{3} - 2$ |
| $2$ | $3$ | $(2,6,3)$ | $\zeta_{3} + 1$ |
| $2$ | $3$ | $(2,3,6)$ | $-\zeta_{3}$ |
| $2$ | $3$ | $(1,4,5)(2,3,6)$ | $-1$ |
| $3$ | $6$ | $(1,2,4,6,5,3)$ | $0$ |
| $3$ | $6$ | $(1,3,5,6,4,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.