Basic invariants
Dimension: | $1$ |
Group: | $C_3$ |
Conductor: | \(63\)\(\medspace = 3^{2} \cdot 7 \) |
Artin field: | Galois closure of 3.3.3969.2 |
Galois orbit size: | $2$ |
Smallest permutation container: | $C_3$ |
Parity: | even |
Dirichlet character: | \(\chi_{63}(16,\cdot)\) |
Projective image: | $C_1$ |
Projective field: | Galois closure of \(\Q\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{3} - 21x - 35 \) . |
The roots of $f$ are computed in $\Q_{ 5 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 3\cdot 5 + 5^{2} + 5^{4} +O(5^{5})\) |
$r_{ 2 }$ | $=$ | \( 1 + 3\cdot 5 + 3\cdot 5^{2} + 5^{3} + 4\cdot 5^{4} +O(5^{5})\) |
$r_{ 3 }$ | $=$ | \( 4 + 3\cdot 5 + 4\cdot 5^{2} + 2\cdot 5^{3} + 4\cdot 5^{4} +O(5^{5})\) |
Generators of the action on the roots $ r_{ 1 }, r_{ 2 }, r_{ 3 } $
Cycle notation |
Character values on conjugacy classes
Size | Order | Action on $ r_{ 1 }, r_{ 2 }, r_{ 3 } $ | Character value | Complex conjugation |
$1$ | $1$ | $()$ | $1$ | ✓ |
$1$ | $3$ | $(1,2,3)$ | $-\zeta_{3} - 1$ | |
$1$ | $3$ | $(1,3,2)$ | $\zeta_{3}$ |