Basic invariants
Dimension: | $1$ |
Group: | $C_4$ |
Conductor: | \(1015\)\(\medspace = 5 \cdot 7 \cdot 29 \) |
Artin field: | Galois closure of 4.4.5151125.2 |
Galois orbit size: | $2$ |
Smallest permutation container: | $C_4$ |
Parity: | even |
Dirichlet character: | \(\chi_{1015}(608,\cdot)\) |
Projective image: | $C_1$ |
Projective field: | Galois closure of \(\Q\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{4} - x^{3} - 254x^{2} + 254x + 12751 \) . |
The roots of $f$ are computed in $\Q_{ 41 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 38\cdot 41 + 35\cdot 41^{2} + 16\cdot 41^{3} + 19\cdot 41^{4} +O(41^{5})\) |
$r_{ 2 }$ | $=$ | \( 1 + 21\cdot 41 + 19\cdot 41^{2} + 9\cdot 41^{3} + 29\cdot 41^{4} +O(41^{5})\) |
$r_{ 3 }$ | $=$ | \( 7 + 25\cdot 41 + 24\cdot 41^{2} + 34\cdot 41^{3} + 8\cdot 41^{4} +O(41^{5})\) |
$r_{ 4 }$ | $=$ | \( 34 + 38\cdot 41 + 41^{2} + 21\cdot 41^{3} + 24\cdot 41^{4} +O(41^{5})\) |
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 |
$1$ | $1$ | $()$ | $1$ |
$1$ | $2$ | $(1,3)(2,4)$ | $-1$ |
$1$ | $4$ | $(1,4,3,2)$ | $\zeta_{4}$ |
$1$ | $4$ | $(1,2,3,4)$ | $-\zeta_{4}$ |
The blue line marks the conjugacy class containing complex conjugation.