Basic invariants
| Dimension: | $1$ |
| Group: | $C_8$ |
| Conductor: | \(113\) |
| Artin field: | Galois closure of 8.8.235260548044817.1 |
| Galois orbit size: | $4$ |
| Smallest permutation container: | $C_8$ |
| Parity: | even |
| Dirichlet character: | \(\chi_{113}(44,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - x^{7} - 49x^{6} - 16x^{5} + 511x^{4} + 367x^{3} - 1499x^{2} - 798x + 1372 \)
|
The roots of $f$ are computed in $\Q_{ 97 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 3 + 80\cdot 97 + 37\cdot 97^{2} + 72\cdot 97^{3} + 43\cdot 97^{4} +O(97^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 10 + 34\cdot 97 + 60\cdot 97^{2} + 67\cdot 97^{3} + 2\cdot 97^{4} +O(97^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 27 + 82\cdot 97 + 89\cdot 97^{2} + 44\cdot 97^{3} + 13\cdot 97^{4} +O(97^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 30 + 40\cdot 97 + 44\cdot 97^{2} + 30\cdot 97^{4} +O(97^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 31 + 39\cdot 97 + 24\cdot 97^{2} + 23\cdot 97^{3} + 76\cdot 97^{4} +O(97^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 55 + 42\cdot 97 + 38\cdot 97^{2} + 11\cdot 97^{3} + 86\cdot 97^{4} +O(97^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 59 + 74\cdot 97 + 29\cdot 97^{2} + 84\cdot 97^{3} + 38\cdot 97^{4} +O(97^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 77 + 91\cdot 97 + 62\cdot 97^{2} + 83\cdot 97^{3} + 96\cdot 97^{4} +O(97^{5})\)
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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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,7)(2,4)(3,8)(5,6)$ | $-1$ |
| $1$ | $4$ | $(1,5,7,6)(2,3,4,8)$ | $-\zeta_{8}^{2}$ |
| $1$ | $4$ | $(1,6,7,5)(2,8,4,3)$ | $\zeta_{8}^{2}$ |
| $1$ | $8$ | $(1,4,5,8,7,2,6,3)$ | $\zeta_{8}^{3}$ |
| $1$ | $8$ | $(1,8,6,4,7,3,5,2)$ | $\zeta_{8}$ |
| $1$ | $8$ | $(1,2,5,3,7,4,6,8)$ | $-\zeta_{8}^{3}$ |
| $1$ | $8$ | $(1,3,6,2,7,8,5,4)$ | $-\zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.