Basic invariants
| Dimension: | $2$ |
| Group: | $S_3\times C_3$ |
| Conductor: | \(57\)\(\medspace = 3 \cdot 19 \) |
| Artin stem field: | Galois closure of 6.0.9747.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $S_3\times C_3$ |
| Parity: | odd |
| Determinant: | 1.57.6t1.a.b |
| Projective image: | $S_3$ |
| Projective stem field: | Galois closure of 3.1.1083.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} + x^{4} - 2x^{3} + 4x^{2} - 3x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$:
\( x^{2} + 7x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 3 a + 2 + \left(2 a + 2\right)\cdot 11 + 7\cdot 11^{2} + \left(a + 1\right)\cdot 11^{3} + \left(10 a + 4\right)\cdot 11^{4} + \left(a + 3\right)\cdot 11^{5} +O(11^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( a + 8 + \left(a + 2\right)\cdot 11^{2} + \left(2 a + 10\right)\cdot 11^{3} + \left(3 a + 2\right)\cdot 11^{4} + \left(9 a + 6\right)\cdot 11^{5} +O(11^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 3 a + 4 + \left(5 a + 7\right)\cdot 11 + 7\cdot 11^{2} + \left(10 a + 6\right)\cdot 11^{3} + \left(6 a + 8\right)\cdot 11^{4} + 10 a\cdot 11^{5} +O(11^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 8 a + 3 + \left(8 a + 8\right)\cdot 11 + \left(10 a + 5\right)\cdot 11^{2} + \left(9 a + 5\right)\cdot 11^{3} + 10\cdot 11^{4} + 9 a\cdot 11^{5} +O(11^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 8 a + 5 + \left(5 a + 3\right)\cdot 11 + \left(10 a + 4\right)\cdot 11^{2} + 2\cdot 11^{3} + \left(4 a + 4\right)\cdot 11^{4} + 3\cdot 11^{5} +O(11^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 10 a + 1 + 10 a\cdot 11 + \left(9 a + 6\right)\cdot 11^{2} + \left(8 a + 6\right)\cdot 11^{3} + \left(7 a + 2\right)\cdot 11^{4} + \left(a + 7\right)\cdot 11^{5} +O(11^{6})\)
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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 |
| $1$ | $1$ | $()$ | $2$ | |
| $3$ | $2$ | $(1,4)(2,6)(3,5)$ | $0$ | ✓ |
| $1$ | $3$ | $(1,2,5)(3,4,6)$ | $2 \zeta_{3}$ | |
| $1$ | $3$ | $(1,5,2)(3,6,4)$ | $-2 \zeta_{3} - 2$ | |
| $2$ | $3$ | $(1,5,2)$ | $-\zeta_{3}$ | |
| $2$ | $3$ | $(1,2,5)$ | $\zeta_{3} + 1$ | |
| $2$ | $3$ | $(1,5,2)(3,4,6)$ | $-1$ | |
| $3$ | $6$ | $(1,3,2,4,5,6)$ | $0$ | |
| $3$ | $6$ | $(1,6,5,4,2,3)$ | $0$ |