Basic invariants
Dimension: | $1$ |
Group: | $C_4$ |
Conductor: | \(35\)\(\medspace = 5 \cdot 7 \) |
Artin number field: | Galois closure of 4.4.6125.1 |
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 $\Q_{ 11 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 6\cdot 11 + 4\cdot 11^{3} + 10\cdot 11^{4} +O(11^{5})\) |
$r_{ 2 }$ | $=$ | \( 1 + 7\cdot 11 + 8\cdot 11^{2} + 2\cdot 11^{3} + 6\cdot 11^{4} +O(11^{5})\) |
$r_{ 3 }$ | $=$ | \( 3 + 7\cdot 11 + 2\cdot 11^{2} + 10\cdot 11^{4} +O(11^{5})\) |
$r_{ 4 }$ | $=$ | \( 8 + 11 + 10\cdot 11^{2} + 3\cdot 11^{3} + 6\cdot 11^{4} +O(11^{5})\) |
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 | |
$c1$ | $c2$ | |||
$1$ | $1$ | $()$ | $1$ | $1$ |
$1$ | $2$ | $(1,4)(2,3)$ | $-1$ | $-1$ |
$1$ | $4$ | $(1,2,4,3)$ | $\zeta_{4}$ | $-\zeta_{4}$ |
$1$ | $4$ | $(1,3,4,2)$ | $-\zeta_{4}$ | $\zeta_{4}$ |