Basic invariants
Dimension: | $1$ |
Group: | $C_4$ |
Conductor: | \(143\)\(\medspace = 11 \cdot 13 \) |
Artin number field: | Galois closure of 4.4.265837.1 |
Galois orbit size: | $2$ |
Smallest permutation container: | $C_4$ |
Parity: | even |
Projective image: | $C_1$ |
Projective field: | Galois closure of \(\Q\) |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 43 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 2 + 29\cdot 43 + 39\cdot 43^{2} + 35\cdot 43^{3} + 23\cdot 43^{4} +O(43^{5})\) |
$r_{ 2 }$ | $=$ | \( 10 + 12\cdot 43 + 43^{2} + 31\cdot 43^{3} +O(43^{5})\) |
$r_{ 3 }$ | $=$ | \( 13 + 16\cdot 43 + 11\cdot 43^{2} + 7\cdot 43^{3} + 3\cdot 43^{4} +O(43^{5})\) |
$r_{ 4 }$ | $=$ | \( 19 + 28\cdot 43 + 33\cdot 43^{2} + 11\cdot 43^{3} + 15\cdot 43^{4} +O(43^{5})\) |
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 | |
$c1$ | $c2$ | |||
$1$ | $1$ | $()$ | $1$ | $1$ |
$1$ | $2$ | $(1,2)(3,4)$ | $-1$ | $-1$ |
$1$ | $4$ | $(1,4,2,3)$ | $\zeta_{4}$ | $-\zeta_{4}$ |
$1$ | $4$ | $(1,3,2,4)$ | $-\zeta_{4}$ | $\zeta_{4}$ |