Normalized defining polynomial
\( x^{4} - 10x^{2} + 4 \)
Invariants
| Degree: | $4$ |
| |
| Signature: | $[4, 0]$ |
| |
| Discriminant: |
\(28224\)
\(\medspace = 2^{6}\cdot 3^{2}\cdot 7^{2}\)
|
| |
| Root discriminant: | \(12.96\) |
| |
| Galois root discriminant: | $2^{3/2}3^{1/2}7^{1/2}\approx 12.96148139681572$ | ||
| Ramified primes: |
\(2\), \(3\), \(7\)
|
| |
| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $C_2^2$ |
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(168=2^{3}\cdot 3\cdot 7\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{168}(1,·)$, $\chi_{168}(139,·)$, $\chi_{168}(155,·)$, $\chi_{168}(41,·)$$\rbrace$ | ||
| This is not a CM field. | |||
| This field has no CM subfields. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{2}a^{2}$, $\frac{1}{2}a^{3}$
| 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: | $3$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-5a+4$, $\frac{1}{2}a^{3}+\frac{1}{2}a^{2}-5a-4$, $a^{3}-\frac{1}{2}a^{2}-10a+6$
|
| |
| Regulator: | \( 12.2123558453 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{0}\cdot 12.2123558453 \cdot 1}{2\cdot\sqrt{28224}}\cr\approx \mathstrut & 0.581540754538 \end{aligned}\]
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
| \(\Q(\sqrt{14}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{21}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Multiplicative Galois module structure
| $U_{K^{gal}}/\textrm{Tors}(U_{K^{gal}}) \cong$ $J$ |
| Galois action is Type III |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | ${\href{/padicField/5.1.0.1}{1} }^{4}$ | R | ${\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.1.0.1}{1} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.2.6a1.1 | $x^{4} + 2 x^{3} + 3 x^{2} + 2 x + 3$ | $2$ | $2$ | $6$ | $C_2^2$ | $$[3]^{2}$$ |
|
\(3\)
| 3.2.2.2a1.2 | $x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ |
|
\(7\)
| 7.2.2.2a1.2 | $x^{4} + 12 x^{3} + 42 x^{2} + 36 x + 16$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ |
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 | 1.24.2t1.a.a | $1$ | $ 2^{3} \cdot 3 $ | \(\Q(\sqrt{6}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| *4 | 1.56.2t1.a.a | $1$ | $ 2^{3} \cdot 7 $ | \(\Q(\sqrt{14}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| *4 | 1.21.2t1.a.a | $1$ | $ 3 \cdot 7 $ | \(\Q(\sqrt{21}) \) | $C_2$ (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 *.