Basic invariants
Dimension: | $1$ |
Group: | $C_4$ |
Conductor: | \(345\)\(\medspace = 3 \cdot 5 \cdot 23 \) |
Artin field: | Galois closure of 4.0.595125.1 |
Galois orbit size: | $2$ |
Smallest permutation container: | $C_4$ |
Parity: | odd |
Dirichlet character: | \(\chi_{345}(68,\cdot)\) |
Projective image: | $C_1$ |
Projective field: | Galois closure of \(\Q\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{4} - x^{3} + 86x^{2} - 86x + 1531 \) . |
The roots of $f$ are computed in $\Q_{ 19 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 4 + 18\cdot 19 + 2\cdot 19^{2} + 18\cdot 19^{3} + 4\cdot 19^{4} +O(19^{5})\) |
$r_{ 2 }$ | $=$ | \( 7 + 16\cdot 19 + 16\cdot 19^{2} + 19^{3} +O(19^{5})\) |
$r_{ 3 }$ | $=$ | \( 11 + 17\cdot 19 + 5\cdot 19^{2} + 5\cdot 19^{3} + 6\cdot 19^{4} +O(19^{5})\) |
$r_{ 4 }$ | $=$ | \( 17 + 4\cdot 19 + 12\cdot 19^{2} + 12\cdot 19^{3} + 7\cdot 19^{4} +O(19^{5})\) |
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.