Basic invariants
| Dimension: | $2$ |
| Group: | $S_3\times C_3$ |
| Conductor: | \(2268\)\(\medspace = 2^{2} \cdot 3^{4} \cdot 7 \) |
| Artin stem field: | Galois closure of 6.0.432081216.3 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $S_3\times C_3$ |
| Parity: | odd |
| Determinant: | 1.252.6t1.j.b |
| Projective image: | $S_3$ |
| Projective stem field: | Galois closure of 3.1.756.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 12x^{4} - 12x^{3} + 36x^{2} + 72x + 120 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$:
\( x^{2} + 49x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 51 a + 10 + \left(39 a + 39\right)\cdot 53 + \left(42 a + 43\right)\cdot 53^{2} + \left(28 a + 23\right)\cdot 53^{3} + \left(18 a + 48\right)\cdot 53^{4} + \left(7 a + 31\right)\cdot 53^{5} +O(53^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 2 a + 2 + \left(13 a + 42\right)\cdot 53 + \left(10 a + 15\right)\cdot 53^{2} + \left(24 a + 43\right)\cdot 53^{3} + \left(34 a + 40\right)\cdot 53^{4} + \left(45 a + 42\right)\cdot 53^{5} +O(53^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 17 a + \left(41 a + 3\right)\cdot 53 + \left(47 a + 20\right)\cdot 53^{2} + \left(40 a + 15\right)\cdot 53^{3} + \left(27 a + 39\right)\cdot 53^{4} + \left(37 a + 1\right)\cdot 53^{5} +O(53^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 36 a + 15 + \left(11 a + 45\right)\cdot 53 + \left(5 a + 10\right)\cdot 53^{2} + \left(12 a + 25\right)\cdot 53^{3} + \left(25 a + 3\right)\cdot 53^{4} + \left(15 a + 18\right)\cdot 53^{5} +O(53^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 15 a + 36 + \left(28 a + 18\right)\cdot 53 + \left(37 a + 26\right)\cdot 53^{2} + \left(16 a + 37\right)\cdot 53^{3} + \left(46 a + 8\right)\cdot 53^{4} + \left(44 a + 45\right)\cdot 53^{5} +O(53^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 38 a + 43 + \left(24 a + 10\right)\cdot 53 + \left(15 a + 42\right)\cdot 53^{2} + \left(36 a + 13\right)\cdot 53^{3} + \left(6 a + 18\right)\cdot 53^{4} + \left(8 a + 19\right)\cdot 53^{5} +O(53^{6})\)
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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 |
| $1$ | $1$ | $()$ | $2$ | |
| $3$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ | ✓ |
| $1$ | $3$ | $(1,6,3)(2,5,4)$ | $-2 \zeta_{3} - 2$ | |
| $1$ | $3$ | $(1,3,6)(2,4,5)$ | $2 \zeta_{3}$ | |
| $2$ | $3$ | $(1,3,6)(2,5,4)$ | $-1$ | |
| $2$ | $3$ | $(2,5,4)$ | $-\zeta_{3}$ | |
| $2$ | $3$ | $(2,4,5)$ | $\zeta_{3} + 1$ | |
| $3$ | $6$ | $(1,5,6,4,3,2)$ | $0$ | |
| $3$ | $6$ | $(1,2,3,4,6,5)$ | $0$ |