Basic invariants
Dimension: | $1$ |
Group: | $C_4$ |
Conductor: | \(116\)\(\medspace = 2^{2} \cdot 29 \) |
Artin number field: | Galois closure of 4.4.390224.1 |
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 $\Q_{ 53 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 12 + 18\cdot 53 + 16\cdot 53^{2} + 35\cdot 53^{4} +O(53^{5})\) |
$r_{ 2 }$ | $=$ | \( 16 + 31\cdot 53 + 34\cdot 53^{3} + 38\cdot 53^{4} +O(53^{5})\) |
$r_{ 3 }$ | $=$ | \( 37 + 21\cdot 53 + 52\cdot 53^{2} + 18\cdot 53^{3} + 14\cdot 53^{4} +O(53^{5})\) |
$r_{ 4 }$ | $=$ | \( 41 + 34\cdot 53 + 36\cdot 53^{2} + 52\cdot 53^{3} + 17\cdot 53^{4} +O(53^{5})\) |
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 | |
$c1$ | $c2$ | |||
$1$ | $1$ | $()$ | $1$ | $1$ |
$1$ | $2$ | $(1,4)(2,3)$ | $-1$ | $-1$ |
$1$ | $4$ | $(1,3,4,2)$ | $\zeta_{4}$ | $-\zeta_{4}$ |
$1$ | $4$ | $(1,2,4,3)$ | $-\zeta_{4}$ | $\zeta_{4}$ |