Basic invariants
Dimension: | $1$ |
Group: | $C_4$ |
Conductor: | \(195\)\(\medspace = 3 \cdot 5 \cdot 13 \) |
Artin field: | Galois closure of 4.4.190125.1 |
Galois orbit size: | $2$ |
Smallest permutation container: | $C_4$ |
Parity: | even |
Dirichlet character: | \(\chi_{195}(38,\cdot)\) |
Projective image: | $C_1$ |
Projective field: | Galois closure of \(\Q\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{4} - x^{3} - 49x^{2} + 49x + 451 \) . |
The roots of $f$ are computed in $\Q_{ 11 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 5\cdot 11 + 5\cdot 11^{3} + 4\cdot 11^{4} +O(11^{5})\) |
$r_{ 2 }$ | $=$ | \( 1 + 2\cdot 11 + 5\cdot 11^{2} + 7\cdot 11^{4} +O(11^{5})\) |
$r_{ 3 }$ | $=$ | \( 4 + 9\cdot 11 + 10\cdot 11^{2} + 8\cdot 11^{3} +O(11^{5})\) |
$r_{ 4 }$ | $=$ | \( 7 + 5\cdot 11 + 5\cdot 11^{2} + 7\cdot 11^{3} + 9\cdot 11^{4} +O(11^{5})\) |
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.