Basic invariants
| Dimension: | $1$ |
| Group: | $C_2$ |
| Conductor: | \(29\) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin field: | Galois closure of \(\Q(\sqrt{29}) \) |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $C_2$ |
| Parity: | even |
| Dirichlet character: | \(\displaystyle\left(\frac{29}{\bullet}\right)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{2} - x - 7 \)
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The roots of $f$ are computed in $\Q_{ 5 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 2 + 2\cdot 5 + 3\cdot 5^{3} + 2\cdot 5^{4} +O(5^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 4 + 2\cdot 5 + 4\cdot 5^{2} + 5^{3} + 2\cdot 5^{4} +O(5^{5})\)
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Generators of the action on the roots $ r_{ 1 }, r_{ 2 } $
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $ r_{ 1 }, r_{ 2 } $ | Character value | Complex conjugation |
| $1$ | $1$ | $()$ | $1$ | ✓ |
| $1$ | $2$ | $(1,2)$ | $-1$ |