Normalized defining polynomial
\( x^{8} + 4x^{6} - 6x^{4} + 76x^{2} + 25 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[0, 4]$ |
| |
| Discriminant: |
\(12230590464\)
\(\medspace = 2^{24}\cdot 3^{6}\)
|
| |
| Root discriminant: | \(18.24\) |
| |
| Galois root discriminant: | $2^{3}3^{3/4}\approx 18.23605645563822$ | ||
| Ramified primes: |
\(2\), \(3\)
|
| |
| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $D_4$ |
| |
| This field is Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | \(\Q(i, \sqrt{6})\) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{8}a^{4}-\frac{1}{2}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a+\frac{3}{8}$, $\frac{1}{40}a^{5}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}+\frac{19}{40}a-\frac{1}{2}$, $\frac{1}{40}a^{6}-\frac{1}{2}a^{3}-\frac{1}{40}a^{2}-\frac{1}{2}a+\frac{1}{4}$, $\frac{1}{80}a^{7}-\frac{1}{80}a^{6}-\frac{1}{80}a^{5}-\frac{1}{16}a^{4}-\frac{11}{80}a^{3}+\frac{31}{80}a^{2}-\frac{9}{80}a-\frac{5}{16}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $5$ |
Class group and class number
| Ideal class group: | $C_{2}$, which has order $2$ |
| |
| Narrow class group: | $C_{2}$, which has order $2$ |
|
Unit group
| Rank: | $3$ |
| |
| Torsion generator: |
\( -\frac{1}{40} a^{7} - \frac{3}{40} a^{5} + \frac{11}{40} a^{3} - \frac{67}{40} a \)
(order $4$)
|
| |
| Fundamental units: |
$\frac{1}{20}a^{7}-\frac{1}{40}a^{6}+\frac{7}{40}a^{5}-\frac{1}{8}a^{4}-\frac{3}{10}a^{3}+\frac{11}{40}a^{2}+\frac{153}{40}a-\frac{17}{8}$, $\frac{1}{5}a^{7}+\frac{1}{8}a^{6}+\frac{29}{40}a^{5}+\frac{5}{8}a^{4}-\frac{39}{20}a^{3}+\frac{5}{8}a^{2}+\frac{511}{40}a+\frac{101}{8}$, $\frac{1}{20}a^{7}-\frac{3}{40}a^{6}+\frac{7}{40}a^{5}-\frac{1}{8}a^{4}-\frac{3}{10}a^{3}+\frac{53}{40}a^{2}-\frac{7}{40}a+\frac{3}{8}$
|
| |
| Regulator: | \( 263.925854443 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 263.925854443 \cdot 2}{4\cdot\sqrt{12230590464}}\cr\approx \mathstrut & 1.85972060053 \end{aligned}\]
Galois group
| A solvable group of order 8 |
| The 5 conjugacy class representatives for $D_4$ |
| Character table for $D_4$ |
Intermediate fields
| \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-6}) \), \(\Q(i, \sqrt{6})\), 4.2.55296.1 x2, 4.0.55296.2 x2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 4 siblings: | 4.2.55296.1, 4.0.55296.2 |
| Minimal sibling: | 4.2.55296.1 |
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 | ${\href{/padicField/5.1.0.1}{1} }^{8}$ | ${\href{/padicField/7.2.0.1}{2} }^{4}$ | ${\href{/padicField/11.2.0.1}{2} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.8.24c1.18 | $x^{8} + 8 x^{5} + 2 x^{4} + 8 x^{3} + 4 x^{2} + 8 x + 10$ | $8$ | $1$ | $24$ | $D_4$ | $$[2, 3, 4]$$ |
|
\(3\)
| 3.2.4.6a1.2 | $x^{8} + 8 x^{7} + 32 x^{6} + 80 x^{5} + 136 x^{4} + 160 x^{3} + 128 x^{2} + 64 x + 19$ | $4$ | $2$ | $6$ | $D_4$ | $$[\ ]_{4}^{2}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| * | 1.24.2t1.b.a | $1$ | $ 2^{3} \cdot 3 $ | \(\Q(\sqrt{-6}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| * | 1.24.2t1.a.a | $1$ | $ 2^{3} \cdot 3 $ | \(\Q(\sqrt{6}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| * | 1.4.2t1.a.a | $1$ | $ 2^{2}$ | \(\Q(\sqrt{-1}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| *2 | 2.2304.4t3.b.a | $2$ | $ 2^{8} \cdot 3^{2}$ | 8.0.12230590464.4 | $D_4$ (as 8T4) | $1$ | $0$ |
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 *.