Basic invariants
| Dimension: | $2$ |
| Group: | $S_3\times C_3$ |
| Conductor: | \(324\)\(\medspace = 2^{2} \cdot 3^{4} \) |
| Artin stem field: | Galois closure of 6.0.419904.3 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $S_3\times C_3$ |
| Parity: | odd |
| Determinant: | 1.36.6t1.b.a |
| Projective image: | $S_3$ |
| Projective stem field: | Galois closure of 3.1.324.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{3} + 9x^{2} - 12x + 5 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 71 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 71 }$:
\( x^{2} + 69x + 7 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 41 a + 48 + \left(44 a + 23\right)\cdot 71 + \left(5 a + 38\right)\cdot 71^{2} + \left(44 a + 52\right)\cdot 71^{3} + \left(62 a + 44\right)\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 59 a + 70 + \left(48 a + 23\right)\cdot 71 + \left(12 a + 4\right)\cdot 71^{2} + \left(a + 40\right)\cdot 71^{3} + \left(46 a + 32\right)\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 30 a + 59 + 26 a\cdot 71 + \left(65 a + 5\right)\cdot 71^{2} + \left(26 a + 64\right)\cdot 71^{3} + \left(8 a + 54\right)\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 12 a + 46 + \left(22 a + 62\right)\cdot 71 + \left(58 a + 51\right)\cdot 71^{2} + \left(69 a + 29\right)\cdot 71^{3} + \left(24 a + 52\right)\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 29 a + 37 + \left(22 a + 7\right)\cdot 71 + \left(18 a + 14\right)\cdot 71^{2} + \left(45 a + 48\right)\cdot 71^{3} + \left(37 a + 34\right)\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 42 a + 24 + \left(48 a + 23\right)\cdot 71 + \left(52 a + 28\right)\cdot 71^{2} + \left(25 a + 49\right)\cdot 71^{3} + \left(33 a + 64\right)\cdot 71^{4} +O(71^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $3$ | $2$ | $(1,5)(2,3)(4,6)$ | $0$ |
| $1$ | $3$ | $(1,2,6)(3,4,5)$ | $-2 \zeta_{3} - 2$ |
| $1$ | $3$ | $(1,6,2)(3,5,4)$ | $2 \zeta_{3}$ |
| $2$ | $3$ | $(3,5,4)$ | $\zeta_{3} + 1$ |
| $2$ | $3$ | $(3,4,5)$ | $-\zeta_{3}$ |
| $2$ | $3$ | $(1,2,6)(3,5,4)$ | $-1$ |
| $3$ | $6$ | $(1,4,2,5,6,3)$ | $0$ |
| $3$ | $6$ | $(1,3,6,5,2,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.