Basic invariants
| Dimension: | $2$ |
| Group: | $S_3\times C_3$ |
| Conductor: | \(315\)\(\medspace = 3^{2} \cdot 5 \cdot 7 \) |
| Artin number field: | Galois closure of 6.0.3472875.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $S_3\times C_3$ |
| Parity: | odd |
| Projective image: | $S_3$ |
| Projective field: | Galois closure of 3.1.2835.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$:
\( x^{2} + 18x + 2 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 14 a + 12 + 16\cdot 19 + \left(17 a + 14\right)\cdot 19^{2} + \left(16 a + 1\right)\cdot 19^{3} + \left(17 a + 2\right)\cdot 19^{4} + \left(9 a + 15\right)\cdot 19^{5} +O(19^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 7 a + 13 + \left(6 a + 5\right)\cdot 19 + \left(9 a + 7\right)\cdot 19^{2} + 13 a\cdot 19^{3} + \left(18 a + 16\right)\cdot 19^{4} + \left(15 a + 10\right)\cdot 19^{5} +O(19^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 17 a + 4 + \left(12 a + 15\right)\cdot 19 + \left(9 a + 7\right)\cdot 19^{2} + \left(13 a + 3\right)\cdot 19^{3} + 8 a\cdot 19^{4} + \left(a + 2\right)\cdot 19^{5} +O(19^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 2 a + 2 + \left(6 a + 11\right)\cdot 19 + \left(9 a + 4\right)\cdot 19^{2} + \left(5 a + 7\right)\cdot 19^{3} + \left(10 a + 14\right)\cdot 19^{4} + \left(17 a + 13\right)\cdot 19^{5} +O(19^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 5 a + 7 + \left(18 a + 3\right)\cdot 19 + \left(a + 12\right)\cdot 19^{2} + \left(2 a + 1\right)\cdot 19^{3} + \left(a + 3\right)\cdot 19^{4} + \left(9 a + 7\right)\cdot 19^{5} +O(19^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 12 a + 1 + \left(12 a + 5\right)\cdot 19 + \left(9 a + 10\right)\cdot 19^{2} + \left(5 a + 4\right)\cdot 19^{3} + 2\cdot 19^{4} + \left(3 a + 8\right)\cdot 19^{5} +O(19^{6})\)
|
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 | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $2$ | $2$ |
| $3$ | $2$ | $(1,2)(3,6)(4,5)$ | $0$ | $0$ |
| $1$ | $3$ | $(1,6,4)(2,3,5)$ | $2 \zeta_{3}$ | $-2 \zeta_{3} - 2$ |
| $1$ | $3$ | $(1,4,6)(2,5,3)$ | $-2 \zeta_{3} - 2$ | $2 \zeta_{3}$ |
| $2$ | $3$ | $(2,3,5)$ | $\zeta_{3} + 1$ | $-\zeta_{3}$ |
| $2$ | $3$ | $(2,5,3)$ | $-\zeta_{3}$ | $\zeta_{3} + 1$ |
| $2$ | $3$ | $(1,6,4)(2,5,3)$ | $-1$ | $-1$ |
| $3$ | $6$ | $(1,5,6,2,4,3)$ | $0$ | $0$ |
| $3$ | $6$ | $(1,3,4,2,6,5)$ | $0$ | $0$ |