Basic invariants
| Dimension: | $2$ |
| Group: | $S_3\times C_3$ |
| Conductor: | \(777\)\(\medspace = 3 \cdot 7 \cdot 37 \) |
| Artin stem field: | Galois closure of 6.0.67013919.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $S_3\times C_3$ |
| Parity: | odd |
| Determinant: | 1.777.6t1.d.a |
| Projective image: | $S_3$ |
| Projective stem field: | Galois closure of 3.1.5439.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} + 10x^{4} + 10x^{3} + 25x^{2} - 9x + 28 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$:
\( x^{2} + 12x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 10 a + 12 + \left(12 a + 11\right)\cdot 13 + \left(4 a + 1\right)\cdot 13^{2} + \left(9 a + 8\right)\cdot 13^{3} + \left(11 a + 3\right)\cdot 13^{4} + \left(5 a + 11\right)\cdot 13^{5} + \left(10 a + 3\right)\cdot 13^{6} + \left(6 a + 12\right)\cdot 13^{7} +O(13^{8})\)
|
| $r_{ 2 }$ | $=$ |
\( 8 a + 6 + \left(11 a + 2\right)\cdot 13 + \left(9 a + 3\right)\cdot 13^{2} + \left(a + 9\right)\cdot 13^{3} + \left(6 a + 4\right)\cdot 13^{4} + \left(4 a + 5\right)\cdot 13^{5} + \left(11 a + 12\right)\cdot 13^{6} + \left(10 a + 5\right)\cdot 13^{7} +O(13^{8})\)
|
| $r_{ 3 }$ | $=$ |
\( 7 a + 9 + \left(8 a + 7\right)\cdot 13 + \left(5 a + 7\right)\cdot 13^{2} + \left(8 a + 2\right)\cdot 13^{3} + \left(12 a + 12\right)\cdot 13^{4} + \left(6 a + 2\right)\cdot 13^{5} + \left(8 a + 3\right)\cdot 13^{6} + \left(4 a + 5\right)\cdot 13^{7} +O(13^{8})\)
|
| $r_{ 4 }$ | $=$ |
\( 3 a + 9 + 13 + \left(8 a + 7\right)\cdot 13^{2} + \left(3 a + 12\right)\cdot 13^{3} + \left(a + 5\right)\cdot 13^{4} + \left(7 a + 5\right)\cdot 13^{5} + \left(2 a + 8\right)\cdot 13^{6} + \left(6 a + 8\right)\cdot 13^{7} +O(13^{8})\)
|
| $r_{ 5 }$ | $=$ |
\( 5 a + 1 + \left(a + 6\right)\cdot 13 + \left(3 a + 1\right)\cdot 13^{2} + \left(11 a + 1\right)\cdot 13^{3} + \left(6 a + 9\right)\cdot 13^{4} + \left(8 a + 3\right)\cdot 13^{5} + \left(a + 6\right)\cdot 13^{6} + \left(2 a + 5\right)\cdot 13^{7} +O(13^{8})\)
|
| $r_{ 6 }$ | $=$ |
\( 6 a + 3 + \left(4 a + 9\right)\cdot 13 + \left(7 a + 4\right)\cdot 13^{2} + \left(4 a + 5\right)\cdot 13^{3} + 3\cdot 13^{4} + \left(6 a + 10\right)\cdot 13^{5} + \left(4 a + 4\right)\cdot 13^{6} + \left(8 a + 1\right)\cdot 13^{7} +O(13^{8})\)
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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 | Complex conjugation |
| $1$ | $1$ | $()$ | $2$ | |
| $3$ | $2$ | $(1,5)(2,6)(3,4)$ | $0$ | ✓ |
| $1$ | $3$ | $(1,2,3)(4,5,6)$ | $2 \zeta_{3}$ | |
| $1$ | $3$ | $(1,3,2)(4,6,5)$ | $-2 \zeta_{3} - 2$ | |
| $2$ | $3$ | $(4,5,6)$ | $\zeta_{3} + 1$ | |
| $2$ | $3$ | $(4,6,5)$ | $-\zeta_{3}$ | |
| $2$ | $3$ | $(1,2,3)(4,6,5)$ | $-1$ | |
| $3$ | $6$ | $(1,5,2,6,3,4)$ | $0$ | |
| $3$ | $6$ | $(1,4,3,6,2,5)$ | $0$ |