Basic invariants
| Dimension: | $2$ |
| Group: | $S_3\times C_3$ |
| Conductor: | \(405\)\(\medspace = 3^{4} \cdot 5 \) |
| Artin stem field: | Galois closure of 6.0.2460375.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $S_3\times C_3$ |
| Parity: | odd |
| Determinant: | 1.45.6t1.b.b |
| Projective image: | $S_3$ |
| Projective stem field: | Galois closure of 3.1.135.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} + 6x^{4} - x^{3} + 9x^{2} - 3x + 4 \)
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The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$:
\( x^{2} + 33x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 14 a + 35 + \left(22 a + 13\right)\cdot 37 + \left(21 a + 1\right)\cdot 37^{2} + \left(32 a + 13\right)\cdot 37^{3} + \left(31 a + 10\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 23 a + 17 + \left(14 a + 15\right)\cdot 37 + \left(15 a + 28\right)\cdot 37^{2} + \left(4 a + 10\right)\cdot 37^{3} + \left(5 a + 31\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 18 a + 33 + 7\cdot 37 + \left(2 a + 25\right)\cdot 37^{2} + \left(17 a + 34\right)\cdot 37^{3} + \left(26 a + 4\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 33 a + 24 + \left(21 a + 13\right)\cdot 37 + \left(19 a + 20\right)\cdot 37^{2} + \left(15 a + 28\right)\cdot 37^{3} + 5 a\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 19 a + 31 + \left(36 a + 28\right)\cdot 37 + \left(34 a + 32\right)\cdot 37^{2} + \left(19 a + 26\right)\cdot 37^{3} + \left(10 a + 19\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 4 a + 8 + \left(15 a + 31\right)\cdot 37 + \left(17 a + 2\right)\cdot 37^{2} + \left(21 a + 34\right)\cdot 37^{3} + \left(31 a + 6\right)\cdot 37^{4} +O(37^{5})\)
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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 | Complex conjugation |
| $1$ | $1$ | $()$ | $2$ | |
| $3$ | $2$ | $(1,4)(2,5)(3,6)$ | $0$ | ✓ |
| $1$ | $3$ | $(1,5,6)(2,3,4)$ | $-2 \zeta_{3} - 2$ | |
| $1$ | $3$ | $(1,6,5)(2,4,3)$ | $2 \zeta_{3}$ | |
| $2$ | $3$ | $(1,6,5)$ | $\zeta_{3} + 1$ | |
| $2$ | $3$ | $(1,5,6)$ | $-\zeta_{3}$ | |
| $2$ | $3$ | $(1,5,6)(2,4,3)$ | $-1$ | |
| $3$ | $6$ | $(1,2,6,4,5,3)$ | $0$ | |
| $3$ | $6$ | $(1,3,5,4,6,2)$ | $0$ |