Normalized defining polynomial
\( x^{4} - 2x^{3} + 77x^{2} - 76x + 2803 \)
Invariants
| Degree: | $4$ |
| |
| Signature: | $(0, 2)$ |
| |
| Discriminant: |
\(1015670545\)
\(\medspace = 5\cdot 59\cdot 151^{3}\)
|
| |
| Root discriminant: | \(178.52\) |
| |
| Galois root discriminant: | $5^{1/2}59^{1/2}151^{3/4}\approx 739.8500686821726$ | ||
| Ramified primes: |
\(5\), \(59\), \(151\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{44545}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $\frac{1}{6}a^{2}-\frac{1}{6}a-\frac{1}{6}$, $\frac{1}{6}a^{3}-\frac{1}{3}a-\frac{1}{6}$
| Monogenic: | No | |
| Index: | $2$ | |
| Inessential primes: | $2$ |
Class group and class number
| Ideal class group: | $C_{14}$, which has order $14$ |
| |
| Narrow class group: | $C_{14}$, which has order $14$ |
|
Unit group
| Rank: | $1$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental unit: |
$\frac{9711006478615}{3}a^{3}+\frac{3827470619605}{6}a^{2}+\frac{603798276798745}{6}a-\frac{13\cdots 53}{2}$
|
| |
| Regulator: | \( 70.20171513763444 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{2}\cdot 70.20171513763444 \cdot 14}{2\cdot\sqrt{1015670545}}\cr\approx \mathstrut & 0.608736118199166 \end{aligned}\]
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 | ${\href{/padicField/2.2.0.1}{2} }^{2}$ | ${\href{/padicField/3.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }$ | ${\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }$ | ${\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.1.0.1}{1} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }$ | ${\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{2}$ | R |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.1.2.1a1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 5.2.1.0a1.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
|
\(59\)
| 59.2.1.0a1.1 | $x^{2} + 58 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 59.1.2.1a1.2 | $x^{2} + 118$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
|
\(151\)
| 151.1.4.3a1.1 | $x^{4} + 151$ | $4$ | $1$ | $3$ | $D_{4}$ | $$[\ ]_{4}^{2}$$ |