Normalized defining polynomial
\( x^{8} - 4x^{7} + 6x^{6} - 4x^{5} - 4x^{4} + 4x^{3} + 4x^{2} - 2 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[4, 2]$ |
| |
| Discriminant: |
\(530841600\)
\(\medspace = 2^{18}\cdot 3^{4}\cdot 5^{2}\)
|
| |
| Root discriminant: | \(12.32\) |
| |
| Galois root discriminant: | $2^{9/4}3^{1/2}5^{1/2}\approx 18.42311740638763$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\)
|
| |
| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| This field has no CM subfields. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{9}a^{7}+\frac{1}{9}a^{6}-\frac{4}{9}a^{5}+\frac{1}{3}a^{4}-\frac{1}{9}a^{3}+\frac{2}{9}a^{2}-\frac{1}{9}a-\frac{2}{9}$
| Monogenic: | Not computed | |
| 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: | $5$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{5}{9}a^{7}-\frac{16}{9}a^{6}+\frac{16}{9}a^{5}-\frac{35}{9}a^{3}+\frac{7}{9}a^{2}+\frac{28}{9}a+\frac{5}{9}$, $\frac{4}{9}a^{7}-\frac{14}{9}a^{6}+\frac{20}{9}a^{5}-\frac{5}{3}a^{4}-\frac{13}{9}a^{3}+\frac{8}{9}a^{2}+\frac{5}{9}a+\frac{1}{9}$, $\frac{5}{9}a^{7}-\frac{16}{9}a^{6}+\frac{16}{9}a^{5}-\frac{35}{9}a^{3}+\frac{7}{9}a^{2}+\frac{19}{9}a+\frac{5}{9}$, $a-1$, $\frac{1}{3}a^{7}-a^{6}+\frac{2}{3}a^{5}+\frac{1}{3}a^{4}-\frac{5}{3}a^{3}+\frac{1}{3}a^{2}+\frac{4}{3}a+1$
|
| |
| Regulator: | \( 32.9840537964 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 32.9840537964 \cdot 1}{2\cdot\sqrt{530841600}}\cr\approx \mathstrut & 0.452138281270 \end{aligned}\]
Galois group
$D_4:C_2^2$ (as 8T22):
| A solvable group of order 32 |
| The 17 conjugacy class representatives for $Q_8:C_2^2$ |
| Character table for $Q_8:C_2^2$ |
Intermediate fields
| \(\Q(\sqrt{6}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{2}, \sqrt{3})\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | data not computed |
| Degree 8 siblings: | data not computed |
| Degree 16 siblings: | 16.0.2174327193600000000.1, 16.0.176120502681600000000.1, 16.0.11007531417600000000.3, 16.0.11007531417600000000.12, 16.0.281792804290560000.2, 16.0.687970713600000000.4, 16.0.176120502681600000000.8, 16.0.176120502681600000000.9, 16.8.176120502681600000000.1 |
| Minimal sibling: | 8.0.51840000.1 |
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 | R | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.8.18b1.9 | $x^{8} + 4 x^{7} + 2 x^{6} + 4 x^{3} + 14$ | $8$ | $1$ | $18$ | $D_4\times C_2$ | $$[2, 2, 3]^{2}$$ |
|
\(3\)
| 3.2.2.2a1.2 | $x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ |
| 3.2.2.2a1.2 | $x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
|
\(5\)
| 5.4.1.0a1.1 | $x^{4} + 4 x^{2} + 4 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ |
| 5.2.2.2a1.1 | $x^{4} + 8 x^{3} + 20 x^{2} + 21 x + 4$ | $2$ | $2$ | $2$ | $C_4$ | $$[\ ]_{2}^{2}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| *32 | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.40.2t1.a.a | $1$ | $ 2^{3} \cdot 5 $ | \(\Q(\sqrt{10}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| 1.40.2t1.b.a | $1$ | $ 2^{3} \cdot 5 $ | \(\Q(\sqrt{-10}) \) | $C_2$ (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) | $1$ | $-1$ | |
| 1.20.2t1.a.a | $1$ | $ 2^{2} \cdot 5 $ | \(\Q(\sqrt{-5}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| *32 | 1.8.2t1.a.a | $1$ | $ 2^{3}$ | \(\Q(\sqrt{2}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| 1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.120.2t1.b.a | $1$ | $ 2^{3} \cdot 3 \cdot 5 $ | \(\Q(\sqrt{-30}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.120.2t1.a.a | $1$ | $ 2^{3} \cdot 3 \cdot 5 $ | \(\Q(\sqrt{30}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| *32 | 1.12.2t1.a.a | $1$ | $ 2^{2} \cdot 3 $ | \(\Q(\sqrt{3}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| *32 | 1.24.2t1.a.a | $1$ | $ 2^{3} \cdot 3 $ | \(\Q(\sqrt{6}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| 1.60.2t1.a.a | $1$ | $ 2^{2} \cdot 3 \cdot 5 $ | \(\Q(\sqrt{15}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| 1.15.2t1.a.a | $1$ | $ 3 \cdot 5 $ | \(\Q(\sqrt{-15}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.24.2t1.b.a | $1$ | $ 2^{3} \cdot 3 $ | \(\Q(\sqrt{-6}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| *32 | 4.230400.8t22.e.a | $4$ | $ 2^{10} \cdot 3^{2} \cdot 5^{2}$ | 8.4.530841600.1 | $Q_8:C_2^2$ (as 8T22) | $1$ | $0$ |
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 *.