Basic invariants
| Dimension: | $2$ |
| Group: | $S_3\times C_3$ |
| Conductor: | \(1260\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \) |
| Artin number field: | Galois closure of 6.0.31752000.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $S_3\times C_3$ |
| Parity: | odd |
| Projective image: | $S_3$ |
| Projective field: | Galois closure of 3.1.79380.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$:
\( x^{2} + 29x + 3 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 19 a + 7 + \left(14 a + 26\right)\cdot 31 + 27\cdot 31^{2} + \left(24 a + 9\right)\cdot 31^{3} + 27 a\cdot 31^{4} + \left(9 a + 1\right)\cdot 31^{5} +O(31^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 5 a + 16 + \left(29 a + 4\right)\cdot 31 + \left(7 a + 12\right)\cdot 31^{2} + \left(27 a + 20\right)\cdot 31^{3} + \left(a + 7\right)\cdot 31^{4} + \left(14 a + 27\right)\cdot 31^{5} +O(31^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 17 a + 29 + \left(14 a + 8\right)\cdot 31 + \left(7 a + 4\right)\cdot 31^{2} + \left(3 a + 16\right)\cdot 31^{3} + \left(5 a + 15\right)\cdot 31^{4} + \left(4 a + 7\right)\cdot 31^{5} +O(31^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 26 a + 26 + \left(a + 26\right)\cdot 31 + \left(23 a + 29\right)\cdot 31^{2} + \left(3 a + 4\right)\cdot 31^{3} + \left(29 a + 15\right)\cdot 31^{4} + \left(16 a + 22\right)\cdot 31^{5} +O(31^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 12 a + 14 + \left(16 a + 5\right)\cdot 31 + \left(30 a + 14\right)\cdot 31^{2} + \left(6 a + 26\right)\cdot 31^{3} + 3 a\cdot 31^{4} + \left(21 a + 24\right)\cdot 31^{5} +O(31^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 14 a + 1 + \left(16 a + 21\right)\cdot 31 + \left(23 a + 4\right)\cdot 31^{2} + \left(27 a + 15\right)\cdot 31^{3} + \left(25 a + 22\right)\cdot 31^{4} + \left(26 a + 10\right)\cdot 31^{5} +O(31^{6})\)
|
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 values | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $2$ | $2$ |
| $3$ | $2$ | $(1,5)(2,4)(3,6)$ | $0$ | $0$ |
| $1$ | $3$ | $(1,4,3)(2,6,5)$ | $2 \zeta_{3}$ | $-2 \zeta_{3} - 2$ |
| $1$ | $3$ | $(1,3,4)(2,5,6)$ | $-2 \zeta_{3} - 2$ | $2 \zeta_{3}$ |
| $2$ | $3$ | $(1,3,4)$ | $-\zeta_{3}$ | $\zeta_{3} + 1$ |
| $2$ | $3$ | $(1,4,3)$ | $\zeta_{3} + 1$ | $-\zeta_{3}$ |
| $2$ | $3$ | $(1,3,4)(2,6,5)$ | $-1$ | $-1$ |
| $3$ | $6$ | $(1,6,4,5,3,2)$ | $0$ | $0$ |
| $3$ | $6$ | $(1,2,3,5,4,6)$ | $0$ | $0$ |