Basic invariants
Dimension: | $1$ |
Group: | $C_4$ |
Conductor: | \(155\)\(\medspace = 5 \cdot 31 \) |
Artin field: | Galois closure of 4.4.120125.1 |
Galois orbit size: | $2$ |
Smallest permutation container: | $C_4$ |
Parity: | even |
Dirichlet character: | \(\chi_{155}(92,\cdot)\) |
Projective image: | $C_1$ |
Projective field: | Galois closure of \(\Q\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{4} - x^{3} - 39x^{2} + 39x + 281 \) . |
The roots of $f$ are computed in $\Q_{ 29 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 10 + 4\cdot 29 + 21\cdot 29^{2} + 9\cdot 29^{3} + 22\cdot 29^{4} +O(29^{5})\) |
$r_{ 2 }$ | $=$ | \( 13 + 21\cdot 29 + 28\cdot 29^{2} + 20\cdot 29^{3} + 19\cdot 29^{4} +O(29^{5})\) |
$r_{ 3 }$ | $=$ | \( 14 + 3\cdot 29 + 3\cdot 29^{2} + 20\cdot 29^{3} + 20\cdot 29^{4} +O(29^{5})\) |
$r_{ 4 }$ | $=$ | \( 22 + 28\cdot 29 + 4\cdot 29^{2} + 7\cdot 29^{3} + 24\cdot 29^{4} +O(29^{5})\) |
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 |
$1$ | $1$ | $()$ | $1$ |
$1$ | $2$ | $(1,3)(2,4)$ | $-1$ |
$1$ | $4$ | $(1,2,3,4)$ | $-\zeta_{4}$ |
$1$ | $4$ | $(1,4,3,2)$ | $\zeta_{4}$ |
The blue line marks the conjugacy class containing complex conjugation.