Normalized defining polynomial
\( x^{4} - 2x^{3} + 4x^{2} - 3x - 59 \)
Invariants
| Degree: | $4$ |
| |
| Signature: | $[2, 1]$ |
| |
| Discriminant: |
\(-23875\)
\(\medspace = -\,5^{3}\cdot 191\)
|
| |
| Root discriminant: | \(12.43\) |
| |
| Galois root discriminant: | $5^{3/4}191^{1/2}\approx 46.210874461670805$ | ||
| Ramified primes: |
\(5\), \(191\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-955}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $\frac{1}{7}a^{2}-\frac{1}{7}a-\frac{2}{7}$, $\frac{1}{7}a^{3}-\frac{3}{7}a-\frac{2}{7}$
| Monogenic: | No | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ |
| |
| Narrow class group: | $C_{2}$, which has order $2$ |
|
Unit group
| Rank: | $2$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{1}{7}a^{2}-\frac{1}{7}a-\frac{2}{7}$, $a^{3}+a^{2}+7a+19$
|
| |
| Regulator: | \( 4.20060067157 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{1}\cdot 4.20060067157 \cdot 1}{2\cdot\sqrt{23875}}\cr\approx \mathstrut & 0.341624942311 \end{aligned}\]
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: | 8.0.20794740015625.1 |
| Degree 4 sibling: | 4.0.4560125.1 |
| 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 | ${\href{/padicField/2.4.0.1}{4} }$ | ${\href{/padicField/3.4.0.1}{4} }$ | R | ${\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }$ | ${\href{/padicField/17.4.0.1}{4} }$ | ${\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }$ | ${\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }$ | ${\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.1.4.3a1.4 | $x^{4} + 20$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ |
|
\(191\)
| 191.1.2.1a1.2 | $x^{2} + 3629$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 191.2.1.0a1.1 | $x^{2} + 190 x + 19$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| * | 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| 1.955.2t1.a.a | $1$ | $ 5 \cdot 191 $ | \(\Q(\sqrt{-955}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.191.2t1.a.a | $1$ | $ 191 $ | \(\Q(\sqrt{-191}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| * | 2.4775.4t3.b.a | $2$ | $ 5^{2} \cdot 191 $ | 4.2.23875.1 | $D_{4}$ (as 4T3) | $1$ | $0$ |
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 *.