Basic invariants
| Dimension: | $1$ |
| Group: | $C_4$ |
| Conductor: | \(10001\)\(\medspace = 73 \cdot 137 \) |
| Artin field: | Galois closure of 4.4.1000300030001.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_4$ |
| Parity: | even |
| Dirichlet character: | \(\chi_{10001}(9079,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} - x^{3} - 3750x^{2} - 79383x - 249439 \)
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The roots of $f$ are computed in $\Q_{ 29 }$ to precision 6.
Roots:
| $r_{ 1 }$ | $=$ |
\( 1 + 21\cdot 29 + 5\cdot 29^{2} + 29^{3} + 20\cdot 29^{4} + 25\cdot 29^{5} +O(29^{6})\)
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| $r_{ 2 }$ | $=$ |
\( 2 + 2\cdot 29 + 5\cdot 29^{2} + 19\cdot 29^{3} + 4\cdot 29^{4} + 20\cdot 29^{5} +O(29^{6})\)
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| $r_{ 3 }$ | $=$ |
\( 7 + 17\cdot 29 + 23\cdot 29^{2} + 10\cdot 29^{3} + 7\cdot 29^{4} + 28\cdot 29^{5} +O(29^{6})\)
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| $r_{ 4 }$ | $=$ |
\( 20 + 17\cdot 29 + 23\cdot 29^{2} + 26\cdot 29^{3} + 25\cdot 29^{4} + 12\cdot 29^{5} +O(29^{6})\)
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Generators of the action on the roots $r_1, \ldots, r_{ 4 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 4 }$ | Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $1$ | $4$ | $(1,4,2,3)$ | $\zeta_{4}$ |
| $1$ | $4$ | $(1,3,2,4)$ | $-\zeta_{4}$ |
The blue line marks the conjugacy class containing complex conjugation.