Normalized defining polynomial
\( x^{2} - x + 185 \)
Invariants
| Degree: | $2$ |
| |
| Signature: | $[0, 1]$ |
| |
| Discriminant: |
\(-739\)
|
| |
| Root discriminant: | \(27.18\) |
| |
| Galois root discriminant: | $739^{1/2}\approx 27.184554438136374$ | ||
| Ramified primes: |
\(739\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-739}) \) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $C_2$ |
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(739\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{739}(1,·)$, $\chi_{739}(738,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | \(\Q(\sqrt{-739}) \) | ||
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: | $C_{5}$, which has order $5$ |
| |
| Narrow class group: | $C_{5}$, which has order $5$ |
| |
| Relative class number: | $5$ |
Unit group
| Rank: | $0$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Regulator: | \( 1 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr =\mathstrut &\frac{2^{0}\cdot(2\pi)^{1}\cdot 1 \cdot 5}{2\cdot\sqrt{739}}\cr\approx \mathstrut & 0.577826769377274 \end{aligned}\]
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 | ${\href{/padicField/2.2.0.1}{2} }$ | ${\href{/padicField/3.2.0.1}{2} }$ | ${\href{/padicField/5.1.0.1}{1} }^{2}$ | ${\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(739\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| * | 1.739.2t1.a.a | $1$ | $ 739 $ | \(\Q(\sqrt{-739}) \) | $C_2$ (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 *.