Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(13\) |
| Artin field: | Galois closure of \(\Q(\zeta_{13})^+\) |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | even |
| Dirichlet character: | \(\chi_{13}(10,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} - 5x^{4} + 4x^{3} + 6x^{2} - 3x - 1 \)
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The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$:
\( x^{2} + 29x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 3 a + 4\cdot 31 + \left(11 a + 23\right)\cdot 31^{2} + 4\cdot 31^{3} + \left(4 a + 10\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 28 a + 6 + \left(30 a + 1\right)\cdot 31 + \left(19 a + 14\right)\cdot 31^{2} + \left(30 a + 25\right)\cdot 31^{3} + \left(26 a + 17\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 24 a + 9 + \left(2 a + 12\right)\cdot 31 + \left(14 a + 18\right)\cdot 31^{2} + \left(27 a + 15\right)\cdot 31^{3} + 15 a\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 18 a + 24 + \left(15 a + 18\right)\cdot 31 + \left(22 a + 28\right)\cdot 31^{2} + \left(23 a + 13\right)\cdot 31^{3} + \left(20 a + 5\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 13 a + 29 + 15 a\cdot 31 + \left(8 a + 27\right)\cdot 31^{2} + \left(7 a + 7\right)\cdot 31^{3} + \left(10 a + 23\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 7 a + 26 + \left(28 a + 24\right)\cdot 31 + \left(16 a + 12\right)\cdot 31^{2} + \left(3 a + 25\right)\cdot 31^{3} + \left(15 a + 4\right)\cdot 31^{4} +O(31^{5})\)
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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 | Complex conjugation |
| $1$ | $1$ | $()$ | $1$ | ✓ |
| $1$ | $2$ | $(1,2)(3,6)(4,5)$ | $-1$ | |
| $1$ | $3$ | $(1,4,3)(2,5,6)$ | $\zeta_{3}$ | |
| $1$ | $3$ | $(1,3,4)(2,6,5)$ | $-\zeta_{3} - 1$ | |
| $1$ | $6$ | $(1,5,3,2,4,6)$ | $-\zeta_{3}$ | |
| $1$ | $6$ | $(1,6,4,2,3,5)$ | $\zeta_{3} + 1$ |