Basic invariants
| Dimension: | $1$ |
| Group: | $C_5$ |
| Conductor: | \(311\) |
| Artin field: | Galois closure of 5.5.9354951841.1 |
| Galois orbit size: | $4$ |
| Smallest permutation container: | $C_5$ |
| Parity: | even |
| Dirichlet character: | \(\chi_{311}(6,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - x^{4} - 124x^{3} - 535x^{2} - 413x + 539 \)
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The roots of $f$ are computed in $\Q_{ 11 }$ to precision 8.
Roots:
| $r_{ 1 }$ | $=$ |
\( 10\cdot 11 + 3\cdot 11^{2} + 3\cdot 11^{3} + 7\cdot 11^{4} + 10\cdot 11^{5} + 3\cdot 11^{6} + 6\cdot 11^{7} +O(11^{8})\)
|
| $r_{ 2 }$ | $=$ |
\( 3 + 7\cdot 11 + 7\cdot 11^{2} + 3\cdot 11^{3} + 7\cdot 11^{4} + 11^{5} + 5\cdot 11^{6} + 3\cdot 11^{7} +O(11^{8})\)
|
| $r_{ 3 }$ | $=$ |
\( 4 + 6\cdot 11 + 3\cdot 11^{2} + 2\cdot 11^{3} + 4\cdot 11^{4} + 7\cdot 11^{5} + 4\cdot 11^{6} + 3\cdot 11^{7} +O(11^{8})\)
|
| $r_{ 4 }$ | $=$ |
\( 6 + 9\cdot 11^{2} + 6\cdot 11^{3} + 3\cdot 11^{4} + 7\cdot 11^{5} + 7\cdot 11^{6} + 3\cdot 11^{7} +O(11^{8})\)
|
| $r_{ 5 }$ | $=$ |
\( 10 + 8\cdot 11 + 8\cdot 11^{2} + 5\cdot 11^{3} + 10\cdot 11^{4} + 5\cdot 11^{5} + 5\cdot 11^{7} +O(11^{8})\)
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Generators of the action on the roots $r_1, \ldots, r_{ 5 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 5 }$ | Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $5$ | $(1,3,4,5,2)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ |
| $1$ | $5$ | $(1,4,2,3,5)$ | $\zeta_{5}^{3}$ |
| $1$ | $5$ | $(1,5,3,2,4)$ | $\zeta_{5}^{2}$ |
| $1$ | $5$ | $(1,2,5,4,3)$ | $\zeta_{5}$ |
The blue line marks the conjugacy class containing complex conjugation.