Basic invariants
| Dimension: | $1$ |
| Group: | $C_4$ |
| Conductor: | \(1740\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 29 \) |
| Artin field: | Galois closure of 4.0.15138000.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_4$ |
| Parity: | odd |
| Dirichlet character: | \(\chi_{1740}(1043,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} + 435x^{2} + 37845 \)
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The roots of $f$ are computed in $\Q_{ 31 }$ to precision 6.
Roots:
| $r_{ 1 }$ | $=$ |
\( 8 + 11\cdot 31 + 10\cdot 31^{2} + 13\cdot 31^{3} + 20\cdot 31^{4} + 21\cdot 31^{5} +O(31^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 11 + 8\cdot 31 + 19\cdot 31^{2} + 29\cdot 31^{3} + 27\cdot 31^{4} + 4\cdot 31^{5} +O(31^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 20 + 22\cdot 31 + 11\cdot 31^{2} + 31^{3} + 3\cdot 31^{4} + 26\cdot 31^{5} +O(31^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 23 + 19\cdot 31 + 20\cdot 31^{2} + 17\cdot 31^{3} + 10\cdot 31^{4} + 9\cdot 31^{5} +O(31^{6})\)
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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 | Complex conjugation |
| $1$ | $1$ | $()$ | $1$ | |
| $1$ | $2$ | $(1,4)(2,3)$ | $-1$ | ✓ |
| $1$ | $4$ | $(1,3,4,2)$ | $-\zeta_{4}$ | |
| $1$ | $4$ | $(1,2,4,3)$ | $\zeta_{4}$ |