Basic invariants
| Dimension: | $1$ |
| Group: | $C_5$ |
| Conductor: | \(275\)\(\medspace = 5^{2} \cdot 11 \) |
| Artin field: | Galois closure of 5.5.5719140625.1 |
| Galois orbit size: | $4$ |
| Smallest permutation container: | $C_5$ |
| Parity: | even |
| Dirichlet character: | \(\chi_{275}(256,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - 110x^{3} - 605x^{2} - 990x - 451 \)
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The roots of $f$ are computed in $\Q_{ 31 }$ to precision 6.
Roots:
| $r_{ 1 }$ | $=$ |
\( 11 + 18\cdot 31 + 16\cdot 31^{2} + 8\cdot 31^{3} + 4\cdot 31^{4} + 27\cdot 31^{5} +O(31^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 17 + 30\cdot 31 + 22\cdot 31^{2} + 13\cdot 31^{3} + 23\cdot 31^{4} + 8\cdot 31^{5} +O(31^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 18 + 20\cdot 31 + 19\cdot 31^{2} + 4\cdot 31^{3} + 8\cdot 31^{4} + 19\cdot 31^{5} +O(31^{6})\)
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| $r_{ 4 }$ | $=$ |
\( 20 + 22\cdot 31 + 16\cdot 31^{2} + 11\cdot 31^{3} + 31^{4} + 24\cdot 31^{5} +O(31^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 27 + 17\cdot 31^{2} + 23\cdot 31^{3} + 24\cdot 31^{4} + 13\cdot 31^{5} +O(31^{6})\)
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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 | Complex conjugation |
| $1$ | $1$ | $()$ | $1$ | ✓ |
| $1$ | $5$ | $(1,2,5,3,4)$ | $\zeta_{5}^{2}$ | |
| $1$ | $5$ | $(1,5,4,2,3)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ | |
| $1$ | $5$ | $(1,3,2,4,5)$ | $\zeta_{5}$ | |
| $1$ | $5$ | $(1,4,3,5,2)$ | $\zeta_{5}^{3}$ |