Normalized defining polynomial
\( x^{2} - x - 51 \)
Invariants
| Degree: | $2$ |
| |
| Signature: | $[2, 0]$ |
| |
| Discriminant: |
\(205\)
\(\medspace = 5\cdot 41\)
|
| |
| Root discriminant: | \(14.32\) |
| |
| Galois root discriminant: | $5^{1/2}41^{1/2}\approx 14.317821063276353$ | ||
| Ramified primes: |
\(5\), \(41\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{205}) \) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $C_2$ |
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(205=5\cdot 41\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{205}(1,·)$, $\chi_{205}(204,·)$$\rbrace$ | ||
| This is not a CM field. | |||
| This field has no CM subfields. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$
| Monogenic: | Yes | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{2}$, which has order $2$ |
| |
| Narrow class group: | $C_{4}$, which has order $4$ |
|
Unit group
| Rank: | $1$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental unit: |
$3a+20$
|
| |
| Regulator: | \( 3.76065884353 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{0}\cdot 3.76065884353 \cdot 2}{2\cdot\sqrt{205}}\cr\approx \mathstrut & 1.05062322735 \end{aligned}\]
Galois group
| A cyclic group of order 2 |
| The 2 conjugacy class representatives for $C_2$ |
| Character table for $C_2$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.2.0.1}{2} }$ | ${\href{/padicField/3.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }$ | R | ${\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.1.0.1}{1} }^{2}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.1.2.1a1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
|
\(41\)
| 41.1.2.1a1.1 | $x^{2} + 41$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| *2 | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| *2 | 1.205.2t1.a.a | $1$ | $ 5 \cdot 41 $ | \(\Q(\sqrt{205}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
Data is given for all irreducible
representations of the Galois group for the Galois closure
of this field. Those marked with * are summands in the
permutation representation coming from this field. Representations
which appear with multiplicity greater than one are indicated
by exponents on the *.