Basic invariants
Dimension: | $1$ |
Group: | $C_3$ |
Conductor: | \(313\) |
Artin field: | Galois closure of 3.3.97969.1 |
Galois orbit size: | $2$ |
Smallest permutation container: | $C_3$ |
Parity: | even |
Dirichlet character: | \(\chi_{313}(214,\cdot)\) |
Projective image: | $C_1$ |
Projective field: | Galois closure of \(\Q\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{3} - x^{2} - 104x - 371 \) . |
The roots of $f$ are computed in $\Q_{ 5 }$ to precision 6.
Roots:
$r_{ 1 }$ | $=$ |
\( 1 + 2\cdot 5^{2} + 3\cdot 5^{4} + 5^{5} +O(5^{6})\)
$r_{ 2 }$ |
$=$ |
\( 2 + 2\cdot 5^{2} + 2\cdot 5^{3} + 3\cdot 5^{4} + 2\cdot 5^{5} +O(5^{6})\)
| $r_{ 3 }$ |
$=$ |
\( 3 + 4\cdot 5 + 2\cdot 5^{3} + 3\cdot 5^{4} +O(5^{6})\)
| |
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 |
$1$ | $1$ | $()$ | $1$ |
$1$ | $3$ | $(1,2,3)$ | $-\zeta_{3} - 1$ |
$1$ | $3$ | $(1,3,2)$ | $\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.