Normalized defining polynomial
\( x^{8} + 15x^{6} + 372x^{4} - 2205x^{2} + 21609 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $(0, 4)$ |
| |
| Discriminant: |
\(3931920170649\)
\(\medspace = 3^{6}\cdot 271^{4}\)
|
| |
| Root discriminant: | \(37.53\) |
| |
| Galois root discriminant: | $3^{3/4}271^{1/2}\approx 37.525422136912695$ | ||
| Ramified primes: |
\(3\), \(271\)
|
| |
| 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: | \(\Q(\sqrt{-3}, \sqrt{-271})\) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{6}a^{4}-\frac{1}{2}a$, $\frac{1}{42}a^{5}-\frac{1}{7}a^{3}-\frac{1}{2}a^{2}-\frac{1}{7}a$, $\frac{1}{54684}a^{6}+\frac{23}{294}a^{4}+\frac{50}{147}a^{2}-\frac{1}{2}a-\frac{5}{124}$, $\frac{1}{765576}a^{7}-\frac{1}{109368}a^{6}-\frac{13}{2058}a^{5}+\frac{13}{294}a^{4}-\frac{391}{2058}a^{3}-\frac{25}{147}a^{2}+\frac{243}{1736}a-\frac{119}{248}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $7$ |
Class group and class number
| Ideal class group: | $C_{11}$, which has order $11$ |
| |
| Narrow class group: | $C_{11}$, which has order $11$ |
|
Unit group
| Rank: | $3$ |
| |
| Torsion generator: |
\( -\frac{5}{18228} a^{6} - \frac{1}{147} a^{4} - \frac{5}{49} a^{2} + \frac{75}{124} \)
(order $6$)
|
| |
| Fundamental units: |
$\frac{17}{31899}a^{7}-\frac{37}{54684}a^{6}-\frac{2}{343}a^{5}-\frac{3}{49}a^{4}+\frac{19}{343}a^{3}+\frac{61}{147}a^{2}-\frac{69}{217}a-\frac{435}{124}$, $\frac{17}{31899}a^{7}+\frac{11}{27342}a^{6}-\frac{2}{343}a^{5}+\frac{8}{147}a^{4}+\frac{19}{343}a^{3}-\frac{76}{147}a^{2}-\frac{69}{217}a+\frac{193}{62}$, $\frac{17}{31899}a^{7}-\frac{37}{54684}a^{6}-\frac{2}{343}a^{5}-\frac{3}{49}a^{4}+\frac{19}{343}a^{3}+\frac{61}{147}a^{2}-\frac{69}{217}a-\frac{311}{124}$
|
| |
| Regulator: | \( 579.746058916 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 579.746058916 \cdot 11}{6\cdot\sqrt{3931920170649}}\cr\approx \mathstrut & 0.835403647843 \end{aligned}\]
Galois group
| A solvable group of order 8 |
| The 5 conjugacy class representatives for $D_4$ |
| Character table for $D_4$ |
Intermediate fields
| \(\Q(\sqrt{-271}) \), \(\Q(\sqrt{813}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-3}, \sqrt{-271})\), 4.2.1982907.1 x2, 4.0.7317.1 x2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 4 siblings: | 4.2.1982907.1, 4.0.7317.1 |
| Minimal sibling: | 4.0.7317.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/5.4.0.1}{4} }^{2}$ | ${\href{/padicField/7.1.0.1}{1} }^{8}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.1.0.1}{1} }^{8}$ | ${\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.2.4.6a1.2 | $x^{8} + 8 x^{7} + 32 x^{6} + 80 x^{5} + 136 x^{4} + 160 x^{3} + 128 x^{2} + 64 x + 19$ | $4$ | $2$ | $6$ | $D_4$ | $$[\ ]_{4}^{2}$$ |
|
\(271\)
| Deg $4$ | $2$ | $2$ | $2$ | |||
| Deg $4$ | $2$ | $2$ | $2$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| *8 | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| *8 | 1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| *8 | 1.813.2t1.a.a | $1$ | $ 3 \cdot 271 $ | \(\Q(\sqrt{813}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| *8 | 1.271.2t1.a.a | $1$ | $ 271 $ | \(\Q(\sqrt{-271}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| *16 | 2.2439.4t3.b.a | $2$ | $ 3^{2} \cdot 271 $ | 8.0.3931920170649.2 | $D_4$ (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 *.