Basic invariants
| Dimension: | $2$ |
| Group: | $S_3\times C_3$ |
| Conductor: | \(3528\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 7^{2} \) |
| Artin stem field: | Galois closure of 6.0.99574272.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $S_3\times C_3$ |
| Parity: | odd |
| Determinant: | 1.504.6t1.q.a |
| Projective image: | $S_3$ |
| Projective stem field: | Galois closure of 3.1.648.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} + 7x^{4} + 16x^{3} + 56x^{2} + 104x + 76 \)
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The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$:
\( x^{2} + 29x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 21 a + 2 + \left(22 a + 21\right)\cdot 31 + \left(22 a + 22\right)\cdot 31^{2} + 2 a\cdot 31^{3} + 11\cdot 31^{4} + \left(8 a + 27\right)\cdot 31^{5} + \left(14 a + 22\right)\cdot 31^{6} +O(31^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 24 a + 1 + \left(8 a + 9\right)\cdot 31 + \left(18 a + 18\right)\cdot 31^{2} + \left(25 a + 22\right)\cdot 31^{3} + \left(21 a + 2\right)\cdot 31^{4} + \left(29 a + 5\right)\cdot 31^{5} + \left(a + 11\right)\cdot 31^{6} +O(31^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 10 a + 13 + \left(8 a + 14\right)\cdot 31 + \left(8 a + 14\right)\cdot 31^{2} + \left(28 a + 14\right)\cdot 31^{3} + \left(30 a + 8\right)\cdot 31^{4} + \left(22 a + 12\right)\cdot 31^{5} + \left(16 a + 12\right)\cdot 31^{6} +O(31^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 27 a + 19 + \left(21 a + 14\right)\cdot 31 + \left(22 a + 30\right)\cdot 31^{2} + \left(15 a + 10\right)\cdot 31^{3} + \left(2 a + 30\right)\cdot 31^{4} + \left(27 a + 7\right)\cdot 31^{5} + \left(9 a + 3\right)\cdot 31^{6} +O(31^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( 7 a + 18 + \left(22 a + 2\right)\cdot 31 + \left(12 a + 15\right)\cdot 31^{2} + \left(5 a + 24\right)\cdot 31^{3} + \left(9 a + 20\right)\cdot 31^{4} + \left(a + 11\right)\cdot 31^{5} + \left(29 a + 16\right)\cdot 31^{6} +O(31^{7})\)
|
| $r_{ 6 }$ | $=$ |
\( 4 a + 11 + 9 a\cdot 31 + \left(8 a + 23\right)\cdot 31^{2} + \left(15 a + 19\right)\cdot 31^{3} + \left(28 a + 19\right)\cdot 31^{4} + \left(3 a + 28\right)\cdot 31^{5} + \left(21 a + 26\right)\cdot 31^{6} +O(31^{7})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $3$ | $2$ | $(1,3)(2,5)(4,6)$ | $0$ |
| $1$ | $3$ | $(1,5,6)(2,4,3)$ | $2 \zeta_{3}$ |
| $1$ | $3$ | $(1,6,5)(2,3,4)$ | $-2 \zeta_{3} - 2$ |
| $2$ | $3$ | $(2,4,3)$ | $\zeta_{3} + 1$ |
| $2$ | $3$ | $(2,3,4)$ | $-\zeta_{3}$ |
| $2$ | $3$ | $(1,6,5)(2,4,3)$ | $-1$ |
| $3$ | $6$ | $(1,3,5,2,6,4)$ | $0$ |
| $3$ | $6$ | $(1,4,6,2,5,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.