Basic invariants
| Dimension: | $2$ |
| Group: | $S_3\times C_3$ |
| Conductor: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Artin number field: | Galois closure of 6.0.41472.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $S_3\times C_3$ |
| Parity: | odd |
| Projective image: | $S_3$ |
| Projective field: | Galois closure of 3.1.648.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$:
\( x^{2} + 33x + 2 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 24 a + 9 + \left(21 a + 32\right)\cdot 37 + \left(4 a + 18\right)\cdot 37^{2} + \left(31 a + 8\right)\cdot 37^{3} + \left(35 a + 15\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 13 a + 31 + \left(15 a + 20\right)\cdot 37 + \left(32 a + 15\right)\cdot 37^{2} + \left(5 a + 17\right)\cdot 37^{3} + \left(a + 16\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 29 a + 18 + \left(35 a + 13\right)\cdot 37 + \left(27 a + 30\right)\cdot 37^{2} + \left(12 a + 29\right)\cdot 37^{3} + 35\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 26 a + 1 + \left(16 a + 12\right)\cdot 37 + \left(12 a + 27\right)\cdot 37^{2} + \left(8 a + 27\right)\cdot 37^{3} + \left(24 a + 20\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 8 a + 23 + \left(a + 16\right)\cdot 37 + \left(9 a + 32\right)\cdot 37^{2} + \left(24 a + 15\right)\cdot 37^{3} + \left(36 a + 24\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 11 a + 31 + \left(20 a + 15\right)\cdot 37 + \left(24 a + 23\right)\cdot 37^{2} + \left(28 a + 11\right)\cdot 37^{3} + \left(12 a + 35\right)\cdot 37^{4} +O(37^{5})\)
|
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 values | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $2$ | $2$ |
| $3$ | $2$ | $(1,2)(3,5)(4,6)$ | $0$ | $0$ |
| $1$ | $3$ | $(1,6,5)(2,4,3)$ | $2 \zeta_{3}$ | $-2 \zeta_{3} - 2$ |
| $1$ | $3$ | $(1,5,6)(2,3,4)$ | $-2 \zeta_{3} - 2$ | $2 \zeta_{3}$ |
| $2$ | $3$ | $(1,5,6)$ | $-\zeta_{3}$ | $\zeta_{3} + 1$ |
| $2$ | $3$ | $(1,6,5)$ | $\zeta_{3} + 1$ | $-\zeta_{3}$ |
| $2$ | $3$ | $(1,5,6)(2,4,3)$ | $-1$ | $-1$ |
| $3$ | $6$ | $(1,3,6,2,5,4)$ | $0$ | $0$ |
| $3$ | $6$ | $(1,4,5,2,6,3)$ | $0$ | $0$ |