Basic invariants
| Dimension: | $1$ |
| Group: | $C_4$ |
| Conductor: | \(10039112\)\(\medspace = 2^{3} \cdot 17 \cdot 97 \cdot 761 \) |
| Artin field: | Galois closure of 4.4.166192436315349056.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_4$ |
| Parity: | even |
| Dirichlet character: | \(\chi_{10039112}(9434877,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} - 2509778x^{2} + 1527952846400 \)
|
The roots of $f$ are computed in $\Q_{ 37 }$ to precision 10.
Roots:
| $r_{ 1 }$ | $=$ |
\( 11 + 3\cdot 37 + 13\cdot 37^{2} + 6\cdot 37^{3} + 29\cdot 37^{4} + 25\cdot 37^{5} + 13\cdot 37^{6} + 14\cdot 37^{7} + 21\cdot 37^{8} + 8\cdot 37^{9} +O(37^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 13 + 36\cdot 37 + 4\cdot 37^{2} + 8\cdot 37^{3} + 37^{4} + 22\cdot 37^{5} + 7\cdot 37^{6} + 5\cdot 37^{7} + 4\cdot 37^{8} + 19\cdot 37^{9} +O(37^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 24 + 32\cdot 37^{2} + 28\cdot 37^{3} + 35\cdot 37^{4} + 14\cdot 37^{5} + 29\cdot 37^{6} + 31\cdot 37^{7} + 32\cdot 37^{8} + 17\cdot 37^{9} +O(37^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 26 + 33\cdot 37 + 23\cdot 37^{2} + 30\cdot 37^{3} + 7\cdot 37^{4} + 11\cdot 37^{5} + 23\cdot 37^{6} + 22\cdot 37^{7} + 15\cdot 37^{8} + 28\cdot 37^{9} +O(37^{10})\)
|
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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,4)(2,3)$ | $-1$ |
| $1$ | $4$ | $(1,3,4,2)$ | $-\zeta_{4}$ |
| $1$ | $4$ | $(1,2,4,3)$ | $\zeta_{4}$ |
The blue line marks the conjugacy class containing complex conjugation.