Basic invariants
| Dimension: | $1$ |
| Group: | $C_8$ |
| Conductor: | \(160\)\(\medspace = 2^{5} \cdot 5 \) |
| Artin field: | Galois closure of 8.0.1342177280000.1 |
| Galois orbit size: | $4$ |
| Smallest permutation container: | $C_8$ |
| Parity: | odd |
| Dirichlet character: | \(\chi_{160}(19,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} + 40x^{6} + 500x^{4} + 2000x^{2} + 1250 \)
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The roots of $f$ are computed in $\Q_{ 79 }$ to precision 6.
Roots:
| $r_{ 1 }$ | $=$ |
\( 2 + 48\cdot 79 + 22\cdot 79^{3} + 43\cdot 79^{4} + 75\cdot 79^{5} +O(79^{6})\)
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| $r_{ 2 }$ | $=$ |
\( 6 + 16\cdot 79 + 67\cdot 79^{2} + 56\cdot 79^{3} + 41\cdot 79^{4} + 6\cdot 79^{5} +O(79^{6})\)
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| $r_{ 3 }$ | $=$ |
\( 23 + 18\cdot 79 + 13\cdot 79^{2} + 54\cdot 79^{3} + 65\cdot 79^{4} + 47\cdot 79^{5} +O(79^{6})\)
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| $r_{ 4 }$ | $=$ |
\( 24 + 9\cdot 79 + 60\cdot 79^{2} + 5\cdot 79^{3} + 64\cdot 79^{4} + 71\cdot 79^{5} +O(79^{6})\)
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| $r_{ 5 }$ | $=$ |
\( 55 + 69\cdot 79 + 18\cdot 79^{2} + 73\cdot 79^{3} + 14\cdot 79^{4} + 7\cdot 79^{5} +O(79^{6})\)
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| $r_{ 6 }$ | $=$ |
\( 56 + 60\cdot 79 + 65\cdot 79^{2} + 24\cdot 79^{3} + 13\cdot 79^{4} + 31\cdot 79^{5} +O(79^{6})\)
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| $r_{ 7 }$ | $=$ |
\( 73 + 62\cdot 79 + 11\cdot 79^{2} + 22\cdot 79^{3} + 37\cdot 79^{4} + 72\cdot 79^{5} +O(79^{6})\)
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| $r_{ 8 }$ | $=$ |
\( 77 + 30\cdot 79 + 78\cdot 79^{2} + 56\cdot 79^{3} + 35\cdot 79^{4} + 3\cdot 79^{5} +O(79^{6})\)
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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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-1$ |
| $1$ | $4$ | $(1,6,8,3)(2,5,7,4)$ | $-\zeta_{8}^{2}$ |
| $1$ | $4$ | $(1,3,8,6)(2,4,7,5)$ | $\zeta_{8}^{2}$ |
| $1$ | $8$ | $(1,2,6,5,8,7,3,4)$ | $\zeta_{8}^{3}$ |
| $1$ | $8$ | $(1,5,3,2,8,4,6,7)$ | $\zeta_{8}$ |
| $1$ | $8$ | $(1,7,6,4,8,2,3,5)$ | $-\zeta_{8}^{3}$ |
| $1$ | $8$ | $(1,4,3,7,8,5,6,2)$ | $-\zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.