Basic invariants
| Dimension: | $2$ |
| Group: | $S_3\times C_3$ |
| Conductor: | \(3800\)\(\medspace = 2^{3} \cdot 5^{2} \cdot 19 \) |
| Artin stem field: | Galois closure of 6.0.115520000.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $S_3\times C_3$ |
| Parity: | odd |
| Determinant: | 1.152.6t1.c.a |
| Projective image: | $S_3$ |
| Projective stem field: | Galois closure of 3.1.72200.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} + 5x^{4} + 10x^{3} + 40x^{2} + 28x + 49 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 18 a + 4 + \left(29 a + 26\right)\cdot 31 + \left(30 a + 10\right)\cdot 31^{2} + \left(30 a + 23\right)\cdot 31^{3} + \left(19 a + 2\right)\cdot 31^{4} + \left(13 a + 20\right)\cdot 31^{5} + \left(12 a + 6\right)\cdot 31^{6} +O(31^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 13 a + 9 + \left(a + 5\right)\cdot 31 + 12\cdot 31^{2} + 23\cdot 31^{3} + \left(11 a + 11\right)\cdot 31^{4} + \left(17 a + 27\right)\cdot 31^{5} + \left(18 a + 17\right)\cdot 31^{6} +O(31^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 24 a + 11 + \left(15 a + 8\right)\cdot 31 + \left(3 a + 14\right)\cdot 31^{2} + \left(19 a + 10\right)\cdot 31^{3} + \left(7 a + 10\right)\cdot 31^{4} + \left(21 a + 6\right)\cdot 31^{5} + \left(29 a + 13\right)\cdot 31^{6} +O(31^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 7 a + 28 + \left(15 a + 15\right)\cdot 31 + \left(27 a + 5\right)\cdot 31^{2} + \left(11 a + 14\right)\cdot 31^{3} + \left(23 a + 6\right)\cdot 31^{4} + \left(9 a + 10\right)\cdot 31^{5} + \left(a + 20\right)\cdot 31^{6} +O(31^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( 6 a + \left(17 a + 20\right)\cdot 31 + \left(3 a + 14\right)\cdot 31^{2} + \left(19 a + 24\right)\cdot 31^{3} + \left(18 a + 21\right)\cdot 31^{4} + 7 a\cdot 31^{5} + \left(17 a + 4\right)\cdot 31^{6} +O(31^{7})\)
|
| $r_{ 6 }$ | $=$ |
\( 25 a + 12 + \left(13 a + 17\right)\cdot 31 + \left(27 a + 4\right)\cdot 31^{2} + \left(11 a + 28\right)\cdot 31^{3} + \left(12 a + 8\right)\cdot 31^{4} + \left(23 a + 28\right)\cdot 31^{5} + \left(13 a + 30\right)\cdot 31^{6} +O(31^{7})\)
|
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 | Complex conjugation |
| $1$ | $1$ | $()$ | $2$ | |
| $3$ | $2$ | $(1,6)(2,4)(3,5)$ | $0$ | ✓ |
| $1$ | $3$ | $(1,5,4)(2,6,3)$ | $2 \zeta_{3}$ | |
| $1$ | $3$ | $(1,4,5)(2,3,6)$ | $-2 \zeta_{3} - 2$ | |
| $2$ | $3$ | $(1,4,5)$ | $-\zeta_{3}$ | |
| $2$ | $3$ | $(1,5,4)$ | $\zeta_{3} + 1$ | |
| $2$ | $3$ | $(1,5,4)(2,3,6)$ | $-1$ | |
| $3$ | $6$ | $(1,3,4,6,5,2)$ | $0$ | |
| $3$ | $6$ | $(1,2,5,6,4,3)$ | $0$ |