Normalized defining polynomial
\( x^{8} - 4x^{7} + 8x^{5} + 14x^{4} - 32x^{3} + 28x^{2} - 48x + 34 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[0, 4]$ |
| |
| Discriminant: |
\(339738624\)
\(\medspace = 2^{22}\cdot 3^{4}\)
|
| |
| Root discriminant: | \(11.65\) |
| |
| Galois root discriminant: | $2^{11/4}3^{1/2}\approx 11.651802520975762$ | ||
| 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: | \(\Q(\zeta_{8})\) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{663}a^{7}-\frac{22}{221}a^{6}-\frac{328}{663}a^{5}-\frac{209}{663}a^{4}+\frac{154}{663}a^{3}-\frac{77}{663}a^{2}-\frac{281}{663}a-\frac{6}{13}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
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: |
\( -\frac{155}{663} a^{7} + \frac{506}{663} a^{6} + \frac{452}{663} a^{5} - \frac{976}{663} a^{4} - \frac{1032}{221} a^{3} + \frac{737}{221} a^{2} - \frac{878}{221} a + \frac{307}{39} \)
(order $8$)
|
| |
| Fundamental units: |
$\frac{437}{663}a^{7}-\frac{1217}{663}a^{6}-\frac{1454}{663}a^{5}+\frac{1708}{663}a^{4}+\frac{2690}{221}a^{3}-\frac{1345}{221}a^{2}+\frac{2310}{221}a-\frac{703}{39}$, $\frac{10}{663}a^{7}+\frac{1}{221}a^{6}+\frac{35}{663}a^{5}-\frac{101}{663}a^{4}-\frac{449}{663}a^{3}-\frac{107}{663}a^{2}+\frac{505}{663}a+\frac{5}{13}$, $\frac{410}{663}a^{7}-\frac{401}{221}a^{6}-\frac{1217}{663}a^{5}+\frac{1826}{663}a^{4}+\frac{7448}{663}a^{3}-\frac{5050}{663}a^{2}+\frac{7445}{663}a-\frac{237}{13}$
|
| |
| Regulator: | \( 29.9370957064 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 29.9370957064 \cdot 2}{8\cdot\sqrt{339738624}}\cr\approx \mathstrut & 0.632844027986 \end{aligned}\]
Galois group
| A solvable group of order 8 |
| The 5 conjugacy class representatives for $D_4$ |
| Character table for $D_4$ |
Intermediate fields
| \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{2}) \), \(\Q(\zeta_{8})\), 4.0.4608.1 x2, 4.2.9216.1 x2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 4 siblings: | 4.2.9216.1, 4.0.4608.1 |
| Minimal sibling: | 4.0.4608.1 |
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 | ${\href{/padicField/5.2.0.1}{2} }^{4}$ | ${\href{/padicField/7.2.0.1}{2} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{4}$ | ${\href{/padicField/17.1.0.1}{1} }^{8}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.8.22d1.10 | $x^{8} + 4 x^{7} + 4 x^{6} + 10 x^{4} + 4 x^{2} + 8 x + 2$ | $8$ | $1$ | $22$ | $D_4$ | $$[2, 3, \frac{7}{2}]$$ |
|
\(3\)
| 3.2.2.2a1.1 | $x^{4} + 4 x^{3} + 8 x^{2} + 11 x + 4$ | $2$ | $2$ | $2$ | $C_4$ | $$[\ ]_{2}^{2}$$ |
| 3.2.2.2a1.1 | $x^{4} + 4 x^{3} + 8 x^{2} + 11 x + 4$ | $2$ | $2$ | $2$ | $C_4$ | $$[\ ]_{2}^{2}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| * | 1.8.2t1.a.a | $1$ | $ 2^{3}$ | \(\Q(\sqrt{2}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| * | 1.4.2t1.a.a | $1$ | $ 2^{2}$ | \(\Q(\sqrt{-1}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| * | 1.8.2t1.b.a | $1$ | $ 2^{3}$ | \(\Q(\sqrt{-2}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| *2 | 2.1152.4t3.i.a | $2$ | $ 2^{7} \cdot 3^{2}$ | 8.0.339738624.8 | $D_4$ (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 *.