Normalized defining polynomial
\( x^{4} - x^{3} + 49x^{2} - 640x - 75020 \)
Invariants
| Degree: | $4$ |
| |
| Signature: | $[2, 1]$ |
| |
| Discriminant: |
\(-30076975\)
\(\medspace = -\,5^{2}\cdot 31\cdot 197^{2}\)
|
| |
| Root discriminant: | \(74.06\) |
| |
| Galois root discriminant: | $5^{1/2}31^{1/2}197^{1/2}\approx 174.7426679434648$ | ||
| Ramified primes: |
\(5\), \(31\), \(197\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-31}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| This field has no CM subfields. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{60664}a^{3}-\frac{247}{60664}a^{2}+\frac{147}{60664}a+\frac{11931}{30332}$
| Monogenic: | No | |
| Index: | $2$ | |
| Inessential primes: | $2$ |
Class group and class number
| Ideal class group: | $C_{2}$, which has order $2$ |
| |
| Narrow class group: | $C_{2}$, which has order $2$ |
|
Unit group
| Rank: | $2$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{1}{60664}a^{3}-\frac{247}{60664}a^{2}+\frac{147}{60664}a+\frac{11931}{30332}$, $\frac{12\cdots 31}{15166}a^{3}+\frac{18\cdots 39}{15166}a^{2}+\frac{37\cdots 43}{15166}a+\frac{27\cdots 08}{7583}$
|
| |
| Regulator: | \( 149.080665214 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{1}\cdot 149.080665214 \cdot 2}{2\cdot\sqrt{30076975}}\cr\approx \mathstrut & 0.683194095368 \end{aligned}\]
Galois group
| A solvable group of order 8 |
| The 5 conjugacy class representatives for $D_{4}$ |
| Character table for $D_{4}$ |
Intermediate fields
| \(\Q(\sqrt{5}) \) |
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: | 4.0.186477245.1 |
| 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 | ${\href{/padicField/2.2.0.1}{2} }^{2}$ | ${\href{/padicField/3.4.0.1}{4} }$ | R | ${\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }$ | ${\href{/padicField/17.4.0.1}{4} }$ | ${\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }$ | ${\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/37.4.0.1}{4} }$ | ${\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }$ | ${\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }$ | ${\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 5.1.2.1a1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
|
\(31\)
| $\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 31.1.2.1a1.1 | $x^{2} + 31$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
|
\(197\)
| 197.2.2.2a1.1 | $x^{4} + 384 x^{3} + 36868 x^{2} + 965 x + 4$ | $2$ | $2$ | $2$ | $C_4$ | $$[\ ]_{2}^{2}$$ |