Normalized defining polynomial
\( x^{4} - 21x^{2} + 16 \)
Invariants
| Degree: | $4$ |
| |
| Signature: | $(4, 0)$ |
| |
| Discriminant: |
\(142129\)
\(\medspace = 13^{2}\cdot 29^{2}\)
|
| |
| Root discriminant: | \(19.42\) |
| |
| Galois root discriminant: | $13^{1/2}29^{1/2}\approx 19.4164878389476$ | ||
| Ramified primes: |
\(13\), \(29\)
|
| |
| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $C_2^2$ |
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(377=13\cdot 29\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{377}(144,·)$, $\chi_{377}(1,·)$, $\chi_{377}(376,·)$, $\chi_{377}(233,·)$$\rbrace$ | ||
| This is not a CM field. | |||
| This field has no CM subfields. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{8}a^{3}-\frac{1}{8}a-\frac{1}{2}$
| Monogenic: | No | |
| Index: | $2$ | |
| Inessential primes: | $2$ |
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: | $3$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{1}{8}a^{3}-\frac{17}{8}a-\frac{3}{2}$, $\frac{1}{8}a^{3}-\frac{25}{8}a-\frac{5}{2}$, $\frac{1}{4}a^{3}-\frac{1}{4}a$
|
| |
| Regulator: | \( 24.1841254988 \) |
| |
| Unit signature rank: | \( 3 \) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{0}\cdot 24.1841254988 \cdot 1}{2\cdot\sqrt{142129}}\cr\approx \mathstrut & 0.513190992017 \end{aligned}\]
Galois group
| An abelian group of order 4 |
| The 4 conjugacy class representatives for $C_2^2$ |
| Character table for $C_2^2$ |
Intermediate fields
| \(\Q(\sqrt{13}) \), \(\Q(\sqrt{29}) \), \(\Q(\sqrt{377}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Multiplicative Galois module structure
| $U_{K^{gal}}/\textrm{Tors}(U_{K^{gal}}) \cong$ $A_1$ $\oplus$ $A_2$ $\oplus$ $A_3$ |
| Galois action is Type I (ii) |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.2.0.1}{2} }^{2}$ | ${\href{/padicField/3.2.0.1}{2} }^{2}$ | ${\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.1.0.1}{1} }^{4}$ | R | ${\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.1.0.1}{1} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(13\)
| 13.1.2.1a1.1 | $x^{2} + 13$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 13.1.2.1a1.1 | $x^{2} + 13$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
|
\(29\)
| 29.1.2.1a1.1 | $x^{2} + 29$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 29.1.2.1a1.1 | $x^{2} + 29$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
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 | 1.13.2t1.a.a | $1$ | $ 13 $ | \(\Q(\sqrt{13}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| *4 | 1.29.2t1.a.a | $1$ | $ 29 $ | \(\Q(\sqrt{29}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| *4 | 1.377.2t1.a.a | $1$ | $ 13 \cdot 29 $ | \(\Q(\sqrt{377}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
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 *.