Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(91\)\(\medspace = 7 \cdot 13 \) |
| Artin field: | Galois closure of 6.6.891474493.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | even |
| Dirichlet character: | \(\chi_{91}(23,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} - 31x^{4} + 4x^{3} + 253x^{2} + 101x - 391 \)
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The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$:
\( x^{2} + 33x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 13 a + 33 + \left(3 a + 27\right)\cdot 37 + \left(20 a + 34\right)\cdot 37^{2} + \left(5 a + 6\right)\cdot 37^{3} + \left(17 a + 9\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 24 a + 11 + \left(33 a + 28\right)\cdot 37 + 16 a\cdot 37^{2} + \left(31 a + 9\right)\cdot 37^{3} + \left(19 a + 35\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 18 a + 10 + \left(18 a + 34\right)\cdot 37 + 18\cdot 37^{2} + \left(35 a + 10\right)\cdot 37^{3} + \left(24 a + 9\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 11 a + 3 + \left(36 a + 9\right)\cdot 37 + \left(14 a + 15\right)\cdot 37^{2} + \left(30 a + 6\right)\cdot 37^{3} + \left(8 a + 26\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 26 a + 10 + 32\cdot 37 + \left(22 a + 1\right)\cdot 37^{2} + \left(6 a + 2\right)\cdot 37^{3} + \left(28 a + 31\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 19 a + 8 + \left(18 a + 16\right)\cdot 37 + \left(36 a + 2\right)\cdot 37^{2} + \left(a + 2\right)\cdot 37^{3} + 12 a\cdot 37^{4} +O(37^{5})\)
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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 |
| $1$ | $1$ | $()$ | $1$ | ✓ |
| $1$ | $2$ | $(1,2)(3,6)(4,5)$ | $-1$ | |
| $1$ | $3$ | $(1,6,5)(2,3,4)$ | $\zeta_{3}$ | |
| $1$ | $3$ | $(1,5,6)(2,4,3)$ | $-\zeta_{3} - 1$ | |
| $1$ | $6$ | $(1,4,6,2,5,3)$ | $\zeta_{3} + 1$ | |
| $1$ | $6$ | $(1,3,5,2,6,4)$ | $-\zeta_{3}$ |