Normalized defining polynomial
\( x^{4} - x^{3} - 32x^{2} + 30x + 123 \)
Invariants
| Degree: | $4$ |
| |
| Signature: | $(4, 0)$ |
| |
| Discriminant: |
\(455877\)
\(\medspace = 3^{2}\cdot 37^{3}\)
|
| |
| Root discriminant: | \(25.98\) |
| |
| Galois root discriminant: | $3^{1/2}37^{3/4}\approx 25.9843537703577$ | ||
| Ramified primes: |
\(3\), \(37\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{37}) \) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $C_4$ |
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(111=3\cdot 37\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{111}(80,·)$, $\chi_{111}(1,·)$, $\chi_{111}(68,·)$, $\chi_{111}(73,·)$$\rbrace$ | ||
| This is not a CM field. | |||
| This field has no CM subfields. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{33}a^{3}-\frac{1}{3}a^{2}+\frac{4}{11}a+\frac{3}{11}$
| Monogenic: | No | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ |
| |
| Narrow class group: | $C_{2}\times C_{2}$, which has order $4$ |
|
Unit group
| Rank: | $3$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{2}{11}a^{3}-\frac{42}{11}a-\frac{59}{11}$, $\frac{3}{11}a^{3}+a^{2}-\frac{41}{11}a-\frac{83}{11}$, $\frac{16}{33}a^{3}+\frac{2}{3}a^{2}-\frac{145}{11}a-\frac{183}{11}$
|
| |
| Regulator: | \( 77.2257721713 \) |
| |
| Unit signature rank: | \( 2 \) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{0}\cdot 77.2257721713 \cdot 1}{2\cdot\sqrt{455877}}\cr\approx \mathstrut & 0.915015404601 \end{aligned}\]
Galois group
| A cyclic group of order 4 |
| The 4 conjugacy class representatives for $C_4$ |
| Character table for $C_4$ |
Intermediate fields
| \(\Q(\sqrt{37}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.4.0.1}{4} }$ | R | ${\href{/padicField/5.4.0.1}{4} }$ | ${\href{/padicField/7.1.0.1}{1} }^{4}$ | ${\href{/padicField/11.1.0.1}{1} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }$ | ${\href{/padicField/17.4.0.1}{4} }$ | ${\href{/padicField/19.4.0.1}{4} }$ | ${\href{/padicField/23.4.0.1}{4} }$ | ${\href{/padicField/29.4.0.1}{4} }$ | ${\href{/padicField/31.4.0.1}{4} }$ | R | ${\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }$ | ${\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.1.2.1a1.2 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 3.1.2.1a1.2 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
|
\(37\)
| 37.1.4.3a1.1 | $x^{4} + 37$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| *4 | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| *4 | 1.111.4t1.a.a | $1$ | $ 3 \cdot 37 $ | \(\Q(\sqrt{222 -6 \sqrt{37}})\) | $C_4$ (as 4T1) | $0$ | $1$ |
| *4 | 1.37.2t1.a.a | $1$ | $ 37 $ | \(\Q(\sqrt{37}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| *4 | 1.111.4t1.a.b | $1$ | $ 3 \cdot 37 $ | \(\Q(\sqrt{222 -6 \sqrt{37}})\) | $C_4$ (as 4T1) | $0$ | $1$ |
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 *.