Basic invariants
| Dimension: | $1$ |
| Group: | $C_4$ |
| Conductor: | \(65\)\(\medspace = 5 \cdot 13 \) |
| Artin field: | Galois closure of 4.4.274625.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_4$ |
| Parity: | even |
| Dirichlet character: | \(\chi_{65}(8,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} - x^{3} - 24x^{2} + 4x + 16 \)
|
The roots of $f$ are computed in $\Q_{ 61 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 12 + 42\cdot 61 + 60\cdot 61^{2} + 58\cdot 61^{3} + 30\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 20 + 11\cdot 61 + 33\cdot 61^{2} + 35\cdot 61^{3} + 11\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 45 + 61 + 11\cdot 61^{2} + 34\cdot 61^{3} + 38\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 46 + 5\cdot 61 + 17\cdot 61^{2} + 54\cdot 61^{3} + 40\cdot 61^{4} +O(61^{5})\)
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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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $1$ | $4$ | $(1,3,2,4)$ | $\zeta_{4}$ |
| $1$ | $4$ | $(1,4,2,3)$ | $-\zeta_{4}$ |
The blue line marks the conjugacy class containing complex conjugation.