Normalized defining polynomial
\( x^{4} + 302x^{2} + 71725 \)
Invariants
| Degree: | $4$ |
| |
| Signature: | $(0, 2)$ |
| |
| Discriminant: |
\(1046657104\)
\(\medspace = 2^{4}\cdot 19\cdot 151^{3}\)
|
| |
| Root discriminant: | \(179.87\) |
| |
| Galois root discriminant: | $2\cdot 19^{1/2}151^{3/4}\approx 375.52556361472006$ | ||
| Ramified primes: |
\(2\), \(19\), \(151\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{2869}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | \(\Q(\sqrt{-151}) \) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{36}a^{2}-\frac{11}{36}$, $\frac{1}{180}a^{3}-\frac{83}{180}a$
| Monogenic: | No | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{2}\times C_{42}$, which has order $84$ |
| |
| Narrow class group: | $C_{2}\times C_{42}$, which has order $84$ |
|
Unit group
| Rank: | $1$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental unit: |
$\frac{87852}{5}a^{3}+\frac{3723845}{6}a^{2}+\frac{50435599}{5}a+\frac{565750637}{6}$
|
| |
| Regulator: | \( 38.86135090892486 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{2}\cdot 38.86135090892486 \cdot 84}{2\cdot\sqrt{1046657104}}\cr\approx \mathstrut & 1.99170359819433 \end{aligned}\]
Galois group
| A solvable group of order 8 |
| The 5 conjugacy class representatives for $D_{4}$ |
| Character table for $D_{4}$ |
Intermediate fields
| \(\Q(\sqrt{-151}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | deg 8 |
| Degree 4 sibling: | deg 4 |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.2.0.1}{2} }^{2}$ | ${\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }$ | ${\href{/padicField/11.1.0.1}{1} }^{4}$ | ${\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.1.0.1}{1} }^{4}$ | R | ${\href{/padicField/23.4.0.1}{4} }$ | ${\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.1.0.1}{1} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.2.2a1.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $$[2]$$ |
| 2.1.2.2a1.2 | $x^{2} + 2 x + 6$ | $2$ | $1$ | $2$ | $C_2$ | $$[2]$$ | |
|
\(19\)
| 19.1.2.1a1.2 | $x^{2} + 38$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 19.2.1.0a1.1 | $x^{2} + 18 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
|
\(151\)
| 151.1.4.3a1.1 | $x^{4} + 151$ | $4$ | $1$ | $3$ | $D_{4}$ | $$[\ ]_{4}^{2}$$ |