Basic invariants
| Dimension: | $1$ |
| Group: | $C_8$ |
| Conductor: | \(137\) |
| Artin field: | Galois closure of 8.0.905824306333433.1 |
| Galois orbit size: | $4$ |
| Smallest permutation container: | $C_8$ |
| Parity: | odd |
| Dirichlet character: | \(\chi_{137}(10,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - x^{7} + 9x^{6} - 105x^{5} + 954x^{4} - 3767x^{3} + 9149x^{2} - 12828x + 7607 \)
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The roots of $f$ are computed in $\Q_{ 73 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 18 + 17\cdot 73 + 64\cdot 73^{3} + 26\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 34 + 40\cdot 73 + 4\cdot 73^{2} + 54\cdot 73^{3} + 19\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 36 + 55\cdot 73 + 41\cdot 73^{2} + 69\cdot 73^{3} + 15\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 45 + 43\cdot 73 + 20\cdot 73^{2} + 39\cdot 73^{3} + 71\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 47 + 4\cdot 73 + 24\cdot 73^{2} + 51\cdot 73^{3} +O(73^{5})\)
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| $r_{ 6 }$ | $=$ |
\( 56 + 24\cdot 73 + 2\cdot 73^{3} + 68\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 64 + 26\cdot 73 + 40\cdot 73^{2} + 48\cdot 73^{3} + 71\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 66 + 5\cdot 73 + 14\cdot 73^{2} + 36\cdot 73^{3} + 17\cdot 73^{4} +O(73^{5})\)
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Generators of the action on the roots $r_1, \ldots, r_{ 8 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 8 }$ | Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,6)(2,4)(3,7)(5,8)$ | $-1$ |
| $1$ | $4$ | $(1,5,6,8)(2,7,4,3)$ | $\zeta_{8}^{2}$ |
| $1$ | $4$ | $(1,8,6,5)(2,3,4,7)$ | $-\zeta_{8}^{2}$ |
| $1$ | $8$ | $(1,7,5,4,6,3,8,2)$ | $-\zeta_{8}$ |
| $1$ | $8$ | $(1,4,8,7,6,2,5,3)$ | $-\zeta_{8}^{3}$ |
| $1$ | $8$ | $(1,3,5,2,6,7,8,4)$ | $\zeta_{8}$ |
| $1$ | $8$ | $(1,2,8,3,6,4,5,7)$ | $\zeta_{8}^{3}$ |
The blue line marks the conjugacy class containing complex conjugation.