Basic invariants
| Dimension: | $1$ |
| Group: | $C_5$ |
| Conductor: | \(421\) |
| Artin field: | Galois closure of 5.5.31414372081.1 |
| Galois orbit size: | $4$ |
| Smallest permutation container: | $C_5$ |
| Parity: | even |
| Dirichlet character: | \(\chi_{421}(252,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - x^{4} - 168x^{3} - 219x^{2} + 3853x + 3517 \)
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The roots of $f$ are computed in $\Q_{ 29 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 2 + 18\cdot 29 + 10\cdot 29^{2} + 14\cdot 29^{4} +O(29^{5})\)
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| $r_{ 2 }$ | $=$ |
\( 7 + 14\cdot 29 + 12\cdot 29^{2} + 25\cdot 29^{3} + 18\cdot 29^{4} +O(29^{5})\)
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| $r_{ 3 }$ | $=$ |
\( 11 + 14\cdot 29 + 11\cdot 29^{2} + 24\cdot 29^{3} + 11\cdot 29^{4} +O(29^{5})\)
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| $r_{ 4 }$ | $=$ |
\( 15 + 18\cdot 29 + 17\cdot 29^{2} + 5\cdot 29^{3} + 16\cdot 29^{4} +O(29^{5})\)
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| $r_{ 5 }$ | $=$ |
\( 24 + 21\cdot 29 + 5\cdot 29^{2} + 2\cdot 29^{3} + 26\cdot 29^{4} +O(29^{5})\)
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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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $5$ | $(1,2,5,3,4)$ | $\zeta_{5}^{2}$ |
| $1$ | $5$ | $(1,5,4,2,3)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ |
| $1$ | $5$ | $(1,3,2,4,5)$ | $\zeta_{5}$ |
| $1$ | $5$ | $(1,4,3,5,2)$ | $\zeta_{5}^{3}$ |
The blue line marks the conjugacy class containing complex conjugation.