Basic invariants
| Dimension: | $1$ |
| Group: | $C_5$ |
| Conductor: | \(41\) |
| Artin field: | Galois closure of 5.5.2825761.1 |
| Galois orbit size: | $4$ |
| Smallest permutation container: | $C_5$ |
| Parity: | even |
| Dirichlet character: | \(\chi_{41}(16,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - x^{4} - 16x^{3} - 5x^{2} + 21x + 9 \)
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The roots of $f$ are computed in $\Q_{ 73 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 41 + 65\cdot 73 + 54\cdot 73^{2} + 73^{3} + 5\cdot 73^{4} +O(73^{5})\)
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| $r_{ 2 }$ | $=$ |
\( 53 + 28\cdot 73 + 29\cdot 73^{2} + 32\cdot 73^{3} + 6\cdot 73^{4} +O(73^{5})\)
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| $r_{ 3 }$ | $=$ |
\( 64 + 69\cdot 73 + 57\cdot 73^{2} + 38\cdot 73^{3} + 42\cdot 73^{4} +O(73^{5})\)
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| $r_{ 4 }$ | $=$ |
\( 67 + 26\cdot 73 + 36\cdot 73^{2} + 59\cdot 73^{3} + 47\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 68 + 27\cdot 73 + 40\cdot 73^{2} + 13\cdot 73^{3} + 44\cdot 73^{4} +O(73^{5})\)
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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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $5$ | $(1,2,3,5,4)$ | $\zeta_{5}$ |
| $1$ | $5$ | $(1,3,4,2,5)$ | $\zeta_{5}^{2}$ |
| $1$ | $5$ | $(1,5,2,4,3)$ | $\zeta_{5}^{3}$ |
| $1$ | $5$ | $(1,4,5,3,2)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ |
The blue line marks the conjugacy class containing complex conjugation.