Normalized defining polynomial
\( x^{2} - 102 \)
Invariants
| Degree: | $2$ |
| |
| Signature: | $[2, 0]$ |
| |
| Discriminant: |
\(408\)
\(\medspace = 2^{3}\cdot 3\cdot 17\)
|
| |
| Root discriminant: | \(20.20\) |
| |
| Galois root discriminant: | $2^{3/2}3^{1/2}17^{1/2}\approx 20.199009876724155$ | ||
| Ramified primes: |
\(2\), \(3\), \(17\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{102}) \) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $C_2$ |
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(408=2^{3}\cdot 3\cdot 17\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{408}(1,·)$, $\chi_{408}(203,·)$$\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: | $C_{2}\times C_{2}$, which has order $4$ |
|
Unit group
| Rank: | $1$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental unit: |
$10a-101$
|
| |
| Regulator: | \( 5.3082431891 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{0}\cdot 5.3082431891 \cdot 2}{2\cdot\sqrt{408}}\cr\approx \mathstrut & 1.0511887902 \end{aligned}\]
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 | R | R | ${\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }$ | R | ${\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.2.3a1.2 | $x^{2} + 10$ | $2$ | $1$ | $3$ | $C_2$ | $$[3]$$ |
|
\(3\)
| 3.1.2.1a1.2 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
|
\(17\)
| 17.1.2.1a1.2 | $x^{2} + 51$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| * | 1.408.2t1.a.a | $1$ | $ 2^{3} \cdot 3 \cdot 17 $ | \(\Q(\sqrt{102}) \) | $C_2$ (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 *.