Basic invariants
| Dimension: | $2$ |
| Group: | $S_3\times C_3$ |
| Conductor: | \(1260\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \) |
| Artin stem field: | Galois closure of 6.0.31752000.4 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $S_3\times C_3$ |
| Parity: | odd |
| Determinant: | 1.1260.6t1.c.a |
| Projective image: | $S_3$ |
| Projective stem field: | Galois closure of 3.1.79380.2 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} + 2x^{4} - 12x^{3} + 21x^{2} + 28x + 56 \)
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The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$:
\( x^{2} + 29x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 7 a + 27 + \left(21 a + 22\right)\cdot 31 + \left(20 a + 19\right)\cdot 31^{2} + \left(21 a + 22\right)\cdot 31^{3} + \left(28 a + 27\right)\cdot 31^{4} + \left(6 a + 24\right)\cdot 31^{5} +O(31^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 9 a + 19 + \left(12 a + 26\right)\cdot 31 + \left(2 a + 26\right)\cdot 31^{2} + \left(14 a + 5\right)\cdot 31^{3} + \left(28 a + 18\right)\cdot 31^{4} + \left(30 a + 8\right)\cdot 31^{5} +O(31^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 2 a + 29 + \left(22 a + 27\right)\cdot 31 + \left(12 a + 22\right)\cdot 31^{2} + \left(23 a + 7\right)\cdot 31^{3} + \left(30 a + 4\right)\cdot 31^{4} + \left(23 a + 26\right)\cdot 31^{5} +O(31^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 24 a + 10 + \left(9 a + 27\right)\cdot 31 + \left(10 a + 8\right)\cdot 31^{2} + \left(9 a + 14\right)\cdot 31^{3} + \left(2 a + 1\right)\cdot 31^{4} + \left(24 a + 10\right)\cdot 31^{5} +O(31^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 22 a + 6 + \left(18 a + 11\right)\cdot 31 + \left(28 a + 19\right)\cdot 31^{2} + 16 a\cdot 31^{3} + \left(2 a + 30\right)\cdot 31^{4} + 10\cdot 31^{5} +O(31^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 29 a + 2 + \left(8 a + 8\right)\cdot 31 + \left(18 a + 26\right)\cdot 31^{2} + \left(7 a + 10\right)\cdot 31^{3} + 11\cdot 31^{4} + \left(7 a + 12\right)\cdot 31^{5} +O(31^{6})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $3$ | $2$ | $(1,6)(2,3)(4,5)$ | $0$ |
| $1$ | $3$ | $(1,3,5)(2,4,6)$ | $2 \zeta_{3}$ |
| $1$ | $3$ | $(1,5,3)(2,6,4)$ | $-2 \zeta_{3} - 2$ |
| $2$ | $3$ | $(1,3,5)(2,6,4)$ | $-1$ |
| $2$ | $3$ | $(2,4,6)$ | $\zeta_{3} + 1$ |
| $2$ | $3$ | $(2,6,4)$ | $-\zeta_{3}$ |
| $3$ | $6$ | $(1,6,3,2,5,4)$ | $0$ |
| $3$ | $6$ | $(1,4,5,2,3,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.