Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(403\)\(\medspace = 13 \cdot 31 \) |
| Artin field: | Galois closure of 6.6.342896882653.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | even |
| Dirichlet character: | \(\chi_{403}(160,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} - 135x^{4} - 74x^{3} + 3672x^{2} + 985x - 10687 \)
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The roots of $f$ are computed in an extension of $\Q_{ 7 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 7 }$:
\( x^{2} + 6x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( a + 4\cdot 7 + \left(a + 1\right)\cdot 7^{2} + 4\cdot 7^{3} + \left(3 a + 6\right)\cdot 7^{4} + \left(5 a + 5\right)\cdot 7^{5} +O(7^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 5 a + 4 + \left(5 a + 3\right)\cdot 7 + \left(2 a + 4\right)\cdot 7^{2} + \left(4 a + 6\right)\cdot 7^{3} + 3\cdot 7^{4} + \left(4 a + 3\right)\cdot 7^{5} +O(7^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 6 a + 1 + \left(6 a + 3\right)\cdot 7 + \left(5 a + 2\right)\cdot 7^{2} + \left(6 a + 3\right)\cdot 7^{3} + \left(3 a + 2\right)\cdot 7^{4} + \left(a + 1\right)\cdot 7^{5} +O(7^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 2 a + 2 + \left(a + 4\right)\cdot 7 + \left(4 a + 1\right)\cdot 7^{2} + \left(2 a + 1\right)\cdot 7^{3} + 6 a\cdot 7^{4} + 2 a\cdot 7^{5} +O(7^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 4 a + 2 + \left(6 a + 5\right)\cdot 7 + \left(3 a + 6\right)\cdot 7^{2} + \left(3 a + 2\right)\cdot 7^{3} + \left(6 a + 2\right)\cdot 7^{4} + 4\cdot 7^{5} +O(7^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 3 a + 6 + \left(3 a + 4\right)\cdot 7^{2} + \left(3 a + 2\right)\cdot 7^{3} + 5\cdot 7^{4} + \left(6 a + 5\right)\cdot 7^{5} +O(7^{6})\)
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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 |
| $1$ | $1$ | $()$ | $1$ | ✓ |
| $1$ | $2$ | $(1,3)(2,4)(5,6)$ | $-1$ | |
| $1$ | $3$ | $(1,2,5)(3,4,6)$ | $-\zeta_{3} - 1$ | |
| $1$ | $3$ | $(1,5,2)(3,6,4)$ | $\zeta_{3}$ | |
| $1$ | $6$ | $(1,4,5,3,2,6)$ | $\zeta_{3} + 1$ | |
| $1$ | $6$ | $(1,6,2,3,5,4)$ | $-\zeta_{3}$ |