Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(315\)\(\medspace = 3^{2} \cdot 5 \cdot 7 \) |
| Artin field: | Galois closure of 6.0.281302875.3 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | odd |
| Dirichlet character: | \(\chi_{315}(34,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 3x^{5} + 24x^{4} - 41x^{3} + 267x^{2} - 306x + 1299 \)
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The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$:
\( x^{2} + 18x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 10 a + 8 + \left(18 a + 1\right)\cdot 19 + \left(a + 3\right)\cdot 19^{2} + \left(10 a + 14\right)\cdot 19^{3} + \left(9 a + 17\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 10 a + 12 + 18 a\cdot 19 + \left(a + 12\right)\cdot 19^{2} + \left(10 a + 3\right)\cdot 19^{3} + \left(9 a + 3\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 9 a + 5 + 3\cdot 19 + \left(17 a + 12\right)\cdot 19^{2} + \left(8 a + 6\right)\cdot 19^{3} + \left(9 a + 17\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 10 a + 14 + \left(18 a + 13\right)\cdot 19 + \left(a + 9\right)\cdot 19^{2} + \left(10 a + 17\right)\cdot 19^{3} + \left(9 a + 17\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 9 a + 18 + 9\cdot 19 + \left(17 a + 5\right)\cdot 19^{2} + \left(8 a + 3\right)\cdot 19^{3} + \left(9 a + 17\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 9 a + 3 + 9\cdot 19 + \left(17 a + 14\right)\cdot 19^{2} + \left(8 a + 11\right)\cdot 19^{3} + \left(9 a + 2\right)\cdot 19^{4} +O(19^{5})\)
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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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,5)(2,6)(3,4)$ | $-1$ |
| $1$ | $3$ | $(1,2,4)(3,5,6)$ | $-\zeta_{3} - 1$ |
| $1$ | $3$ | $(1,4,2)(3,6,5)$ | $\zeta_{3}$ |
| $1$ | $6$ | $(1,6,4,5,2,3)$ | $\zeta_{3} + 1$ |
| $1$ | $6$ | $(1,3,2,5,4,6)$ | $-\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.