Basic invariants
Dimension: | $1$ |
Group: | $C_4$ |
Conductor: | \(113\) |
Artin field: | Galois closure of 4.4.1442897.1 |
Galois orbit size: | $2$ |
Smallest permutation container: | $C_4$ |
Parity: | even |
Dirichlet character: | \(\chi_{113}(15,\cdot)\) |
Projective image: | $C_1$ |
Projective field: | Galois closure of \(\Q\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{4} - x^{3} - 42x^{2} + 120x - 64 \) . |
The roots of $f$ are computed in $\Q_{ 7 }$ to precision 6.
Roots:
$r_{ 1 }$ | $=$ | \( 1 + 6\cdot 7 + 5\cdot 7^{3} + 4\cdot 7^{4} + 5\cdot 7^{5} +O(7^{6})\) |
$r_{ 2 }$ | $=$ | \( 3 + 5\cdot 7 + 5\cdot 7^{2} + 5\cdot 7^{3} + 7^{5} +O(7^{6})\) |
$r_{ 3 }$ | $=$ | \( 5 + 5\cdot 7 + 6\cdot 7^{2} + 4\cdot 7^{4} +O(7^{6})\) |
$r_{ 4 }$ | $=$ | \( 6 + 3\cdot 7 + 2\cdot 7^{3} + 4\cdot 7^{4} + 6\cdot 7^{5} +O(7^{6})\) |
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 |
$1$ | $1$ | $()$ | $1$ |
$1$ | $2$ | $(1,4)(2,3)$ | $-1$ |
$1$ | $4$ | $(1,2,4,3)$ | $\zeta_{4}$ |
$1$ | $4$ | $(1,3,4,2)$ | $-\zeta_{4}$ |
The blue line marks the conjugacy class containing complex conjugation.