Basic invariants
| Dimension: | $1$ |
| Group: | $C_4$ |
| Conductor: | \(240\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \) |
| Artin field: | Galois closure of 4.0.2304000.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_4$ |
| Parity: | odd |
| Dirichlet character: | \(\chi_{240}(227,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} + 60x^{2} + 810 \)
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The roots of $f$ are computed in $\Q_{ 13 }$ to precision 6.
Roots:
| $r_{ 1 }$ | $=$ |
\( 1 + 9\cdot 13 + 7\cdot 13^{2} + 5\cdot 13^{3} + 6\cdot 13^{4} + 9\cdot 13^{5} +O(13^{6})\)
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| $r_{ 2 }$ | $=$ |
\( 2 + 4\cdot 13 + 4\cdot 13^{2} + 13^{3} + 8\cdot 13^{4} + 10\cdot 13^{5} +O(13^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 11 + 8\cdot 13 + 8\cdot 13^{2} + 11\cdot 13^{3} + 4\cdot 13^{4} + 2\cdot 13^{5} +O(13^{6})\)
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| $r_{ 4 }$ | $=$ |
\( 12 + 3\cdot 13 + 5\cdot 13^{2} + 7\cdot 13^{3} + 6\cdot 13^{4} + 3\cdot 13^{5} +O(13^{6})\)
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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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,4)(2,3)$ | $-1$ |
| $1$ | $4$ | $(1,3,4,2)$ | $\zeta_{4}$ |
| $1$ | $4$ | $(1,2,4,3)$ | $-\zeta_{4}$ |
The blue line marks the conjugacy class containing complex conjugation.