Basic invariants
| Dimension: | $1$ |
| Group: | $C_4$ |
| Conductor: | \(760\)\(\medspace = 2^{3} \cdot 5 \cdot 19 \) |
| Artin field: | Galois closure of 4.4.2888000.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_4$ |
| Parity: | even |
| Dirichlet character: | \(\chi_{760}(37,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} - 190x^{2} + 7220 \)
|
The roots of $f$ are computed in $\Q_{ 79 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 1 + 66\cdot 79 + 44\cdot 79^{2} + 41\cdot 79^{3} + 39\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 30 + 49\cdot 79 + 38\cdot 79^{2} + 9\cdot 79^{3} + 17\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 49 + 29\cdot 79 + 40\cdot 79^{2} + 69\cdot 79^{3} + 61\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 78 + 12\cdot 79 + 34\cdot 79^{2} + 37\cdot 79^{3} + 39\cdot 79^{4} +O(79^{5})\)
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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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,4)(2,3)$ | $-1$ |
| $1$ | $4$ | $(1,3,4,2)$ | $\zeta_{4}$ |
| $1$ | $4$ | $(1,2,4,3)$ | $-\zeta_{4}$ |
The blue line marks the conjugacy class containing complex conjugation.