Basic invariants
Dimension: | $1$ |
Group: | $C_4$ |
Conductor: | \(1005\)\(\medspace = 3 \cdot 5 \cdot 67 \) |
Artin number field: | Galois closure of 4.0.5050125.1 |
Galois orbit size: | $2$ |
Smallest permutation container: | $C_4$ |
Parity: | odd |
Projective image: | $C_1$ |
Projective field: | Galois closure of \(\Q\) |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 29 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 10 + 8\cdot 29 + 22\cdot 29^{2} + 10\cdot 29^{3} + 11\cdot 29^{4} +O(29^{5})\) |
$r_{ 2 }$ | $=$ | \( 13 + 29 + 26\cdot 29^{2} + 4\cdot 29^{3} + 23\cdot 29^{4} +O(29^{5})\) |
$r_{ 3 }$ | $=$ | \( 14 + 28\cdot 29 + 29^{2} + 19\cdot 29^{3} + 2\cdot 29^{4} +O(29^{5})\) |
$r_{ 4 }$ | $=$ | \( 22 + 19\cdot 29 + 7\cdot 29^{2} + 23\cdot 29^{3} + 20\cdot 29^{4} +O(29^{5})\) |
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 values | |
$c1$ | $c2$ | |||
$1$ | $1$ | $()$ | $1$ | $1$ |
$1$ | $2$ | $(1,3)(2,4)$ | $-1$ | $-1$ |
$1$ | $4$ | $(1,4,3,2)$ | $\zeta_{4}$ | $-\zeta_{4}$ |
$1$ | $4$ | $(1,2,3,4)$ | $-\zeta_{4}$ | $\zeta_{4}$ |