Basic invariants
Dimension: | $1$ |
Group: | $C_4$ |
Conductor: | \(65\)\(\medspace = 5 \cdot 13 \) |
Artin number field: | Galois closure of 4.0.54925.1 |
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 $\Q_{ 17 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ |
\( 6 + 16\cdot 17 + 6\cdot 17^{2} + 2\cdot 17^{3} + 2\cdot 17^{4} +O(17^{5})\)
$r_{ 2 }$ |
$=$ |
\( 7 + 2\cdot 17 + 16\cdot 17^{2} + 5\cdot 17^{3} + 9\cdot 17^{4} +O(17^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 8 + 16\cdot 17 + 16\cdot 17^{3} + 12\cdot 17^{4} +O(17^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 14 + 15\cdot 17 + 9\cdot 17^{2} + 9\cdot 17^{3} + 9\cdot 17^{4} +O(17^{5})\)
| |
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 | |
$c1$ | $c2$ | |||
$1$ | $1$ | $()$ | $1$ | $1$ |
$1$ | $2$ | $(1,2)(3,4)$ | $-1$ | $-1$ |
$1$ | $4$ | $(1,3,2,4)$ | $\zeta_{4}$ | $-\zeta_{4}$ |
$1$ | $4$ | $(1,4,2,3)$ | $-\zeta_{4}$ | $\zeta_{4}$ |