Basic invariants
| Dimension: | $2$ |
| Group: | $S_3\times C_3$ |
| Conductor: | \(3447\)\(\medspace = 3^{2} \cdot 383 \) |
| Artin number field: | Galois closure of 6.0.4550732847.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $S_3\times C_3$ |
| Parity: | odd |
| Projective image: | $S_3$ |
| Projective field: | Galois closure of 3.1.31023.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$:
\( x^{2} + 33x + 2 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 10 a + 1 + \left(26 a + 1\right)\cdot 37 + \left(2 a + 17\right)\cdot 37^{2} + \left(12 a + 20\right)\cdot 37^{3} + \left(23 a + 11\right)\cdot 37^{4} + \left(22 a + 20\right)\cdot 37^{5} + \left(a + 5\right)\cdot 37^{6} +O(37^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 13 a + 23 + \left(3 a + 30\right)\cdot 37 + \left(5 a + 8\right)\cdot 37^{2} + \left(2 a + 28\right)\cdot 37^{3} + \left(8 a + 4\right)\cdot 37^{4} + \left(32 a + 32\right)\cdot 37^{5} + \left(26 a + 28\right)\cdot 37^{6} +O(37^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 23 a + 32 + \left(29 a + 20\right)\cdot 37 + \left(7 a + 9\right)\cdot 37^{2} + \left(14 a + 13\right)\cdot 37^{3} + \left(31 a + 20\right)\cdot 37^{4} + \left(17 a + 18\right)\cdot 37^{5} + \left(28 a + 17\right)\cdot 37^{6} +O(37^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 14 a + 13 + \left(7 a + 5\right)\cdot 37 + \left(29 a + 11\right)\cdot 37^{2} + \left(22 a + 25\right)\cdot 37^{3} + \left(5 a + 20\right)\cdot 37^{4} + \left(19 a + 21\right)\cdot 37^{5} + \left(8 a + 2\right)\cdot 37^{6} +O(37^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( 24 a + 1 + \left(33 a + 31\right)\cdot 37 + \left(31 a + 25\right)\cdot 37^{2} + \left(34 a + 31\right)\cdot 37^{3} + \left(28 a + 34\right)\cdot 37^{4} + \left(4 a + 4\right)\cdot 37^{5} + \left(10 a + 30\right)\cdot 37^{6} +O(37^{7})\)
|
| $r_{ 6 }$ | $=$ |
\( 27 a + 4 + \left(10 a + 22\right)\cdot 37 + \left(34 a + 1\right)\cdot 37^{2} + \left(24 a + 29\right)\cdot 37^{3} + \left(13 a + 18\right)\cdot 37^{4} + \left(14 a + 13\right)\cdot 37^{5} + \left(35 a + 26\right)\cdot 37^{6} +O(37^{7})\)
|
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 values | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $2$ | $2$ |
| $3$ | $2$ | $(1,3)(2,6)(4,5)$ | $0$ | $0$ |
| $1$ | $3$ | $(1,2,4)(3,6,5)$ | $2 \zeta_{3}$ | $-2 \zeta_{3} - 2$ |
| $1$ | $3$ | $(1,4,2)(3,5,6)$ | $-2 \zeta_{3} - 2$ | $2 \zeta_{3}$ |
| $2$ | $3$ | $(1,4,2)$ | $-\zeta_{3}$ | $\zeta_{3} + 1$ |
| $2$ | $3$ | $(1,2,4)$ | $\zeta_{3} + 1$ | $-\zeta_{3}$ |
| $2$ | $3$ | $(1,4,2)(3,6,5)$ | $-1$ | $-1$ |
| $3$ | $6$ | $(1,6,2,5,4,3)$ | $0$ | $0$ |
| $3$ | $6$ | $(1,3,4,5,2,6)$ | $0$ | $0$ |