Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(777\)\(\medspace = 3 \cdot 7 \cdot 37 \) |
| Artin field: | Galois closure of 6.0.121496235147.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | odd |
| Dirichlet character: | \(\chi_{777}(158,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} + 87x^{4} + 182x^{3} + 7348x^{2} + 4128x + 2304 \)
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The roots of $f$ are computed in an extension of $\Q_{ 89 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 89 }$:
\( x^{2} + 82x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 4 a + 16 + \left(6 a + 38\right)\cdot 89 + \left(a + 62\right)\cdot 89^{2} + \left(32 a + 61\right)\cdot 89^{3} + \left(39 a + 35\right)\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 47 a + 17 + \left(66 a + 85\right)\cdot 89 + \left(46 a + 63\right)\cdot 89^{2} + \left(64 a + 60\right)\cdot 89^{3} + \left(58 a + 18\right)\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 85 a + 44 + \left(82 a + 76\right)\cdot 89 + \left(87 a + 63\right)\cdot 89^{2} + \left(56 a + 17\right)\cdot 89^{3} + \left(49 a + 12\right)\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 49 a + 18 + \left(16 a + 15\right)\cdot 89 + \left(26 a + 15\right)\cdot 89^{2} + \left(66 a + 12\right)\cdot 89^{3} + \left(14 a + 77\right)\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 42 a + 79 + \left(22 a + 58\right)\cdot 89 + \left(42 a + 57\right)\cdot 89^{2} + \left(24 a + 20\right)\cdot 89^{3} + \left(30 a + 9\right)\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 40 a + 5 + \left(72 a + 82\right)\cdot 89 + \left(62 a + 3\right)\cdot 89^{2} + \left(22 a + 5\right)\cdot 89^{3} + \left(74 a + 25\right)\cdot 89^{4} +O(89^{5})\)
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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 |
| $1$ | $1$ | $()$ | $1$ | |
| $1$ | $2$ | $(1,3)(2,5)(4,6)$ | $-1$ | ✓ |
| $1$ | $3$ | $(1,5,4)(2,6,3)$ | $-\zeta_{3} - 1$ | |
| $1$ | $3$ | $(1,4,5)(2,3,6)$ | $\zeta_{3}$ | |
| $1$ | $6$ | $(1,6,5,3,4,2)$ | $-\zeta_{3}$ | |
| $1$ | $6$ | $(1,2,4,3,5,6)$ | $\zeta_{3} + 1$ |