Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(7\) |
| Artin number field: | Galois closure of \(\Q(\zeta_{7})\) |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | odd |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$:
\( x^{2} + 12x + 2 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 6 a + 6 + \left(9 a + 3\right)\cdot 13 + \left(2 a + 5\right)\cdot 13^{2} + \left(3 a + 10\right)\cdot 13^{3} + \left(10 a + 2\right)\cdot 13^{4} +O(13^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( a + 9 + \left(8 a + 9\right)\cdot 13 + 10 a\cdot 13^{2} + \left(a + 4\right)\cdot 13^{3} + \left(2 a + 7\right)\cdot 13^{4} +O(13^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 4 a + 6 + \left(3 a + 1\right)\cdot 13 + \left(9 a + 6\right)\cdot 13^{2} + \left(2 a + 12\right)\cdot 13^{3} + \left(5 a + 10\right)\cdot 13^{4} +O(13^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 9 a + 10 + 9 a\cdot 13 + \left(3 a + 12\right)\cdot 13^{2} + \left(10 a + 5\right)\cdot 13^{3} + 7 a\cdot 13^{4} +O(13^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 12 a + 10 + \left(4 a + 3\right)\cdot 13 + \left(2 a + 3\right)\cdot 13^{2} + \left(11 a + 8\right)\cdot 13^{3} + \left(10 a + 7\right)\cdot 13^{4} +O(13^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 7 a + 12 + \left(3 a + 6\right)\cdot 13 + \left(10 a + 11\right)\cdot 13^{2} + \left(9 a + 10\right)\cdot 13^{3} + \left(2 a + 9\right)\cdot 13^{4} +O(13^{5})\)
|
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 | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $1$ | $1$ |
| $1$ | $2$ | $(1,6)(2,5)(3,4)$ | $-1$ | $-1$ |
| $1$ | $3$ | $(1,4,5)(2,6,3)$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ |
| $1$ | $3$ | $(1,5,4)(2,3,6)$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ |
| $1$ | $6$ | $(1,2,4,6,5,3)$ | $\zeta_{3} + 1$ | $-\zeta_{3}$ |
| $1$ | $6$ | $(1,3,5,6,4,2)$ | $-\zeta_{3}$ | $\zeta_{3} + 1$ |