Normalized defining polynomial
\( x^{8} + 7x^{6} + 76x^{4} - 189x^{2} + 729 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[0, 4]$ |
| |
| Discriminant: |
\(49213429281\)
\(\medspace = 3^{4}\cdot 157^{4}\)
|
| |
| Root discriminant: | \(21.70\) |
| |
| Galois root discriminant: | $3^{1/2}157^{1/2}\approx 21.702534414210707$ | ||
| Ramified primes: |
\(3\), \(157\)
|
| |
| 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: | \(\Q(\sqrt{-3}, \sqrt{157})\) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{6}a^{5}+\frac{1}{6}a^{3}-\frac{1}{2}a^{2}-\frac{1}{3}a-\frac{1}{2}$, $\frac{1}{2052}a^{6}+\frac{4}{27}a^{4}+\frac{1}{27}a^{2}-\frac{7}{76}$, $\frac{1}{36936}a^{7}-\frac{1}{4104}a^{6}+\frac{2}{243}a^{5}-\frac{2}{27}a^{4}-\frac{13}{243}a^{3}-\frac{1}{54}a^{2}-\frac{539}{1368}a+\frac{7}{152}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $3$ |
Class group and class number
| Ideal class group: | $C_{4}$, which has order $4$ |
| |
| Narrow class group: | $C_{4}$, which has order $4$ |
|
Unit group
| Rank: | $3$ |
| |
| Torsion generator: |
\( -\frac{7}{2052} a^{6} - \frac{1}{27} a^{4} - \frac{7}{27} a^{2} + \frac{49}{76} \)
(order $6$)
|
| |
| Fundamental units: |
$\frac{10}{4617}a^{7}-\frac{7}{1026}a^{6}-\frac{2}{243}a^{5}-\frac{2}{27}a^{4}+\frac{13}{243}a^{3}-\frac{14}{27}a^{2}-\frac{374}{171}a-\frac{103}{38}$, $\frac{83}{9234}a^{7}-\frac{7}{1026}a^{6}+\frac{16}{243}a^{5}-\frac{2}{27}a^{4}+\frac{139}{243}a^{3}-\frac{14}{27}a^{2}-\frac{505}{342}a+\frac{125}{38}$, $\frac{1}{57}a^{7}-\frac{31}{2052}a^{6}+\frac{11}{27}a^{4}-\frac{31}{27}a^{2}+\frac{20}{57}a+\frac{217}{76}$
|
| |
| Regulator: | \( 324.950191435 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 324.950191435 \cdot 4}{6\cdot\sqrt{49213429281}}\cr\approx \mathstrut & 1.52195985552 \end{aligned}\]
Galois group
| A solvable group of order 8 |
| The 5 conjugacy class representatives for $D_4$ |
| Character table for $D_4$ |
Intermediate fields
| \(\Q(\sqrt{-471}) \), \(\Q(\sqrt{157}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-3}, \sqrt{157})\), 4.2.73947.1 x2, 4.0.1413.1 x2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 4 siblings: | 4.2.73947.1, 4.0.1413.1 |
| Minimal sibling: | 4.0.1413.1 |
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 | ${\href{/padicField/5.4.0.1}{4} }^{2}$ | ${\href{/padicField/7.2.0.1}{2} }^{4}$ | ${\href{/padicField/11.2.0.1}{2} }^{4}$ | ${\href{/padicField/13.1.0.1}{1} }^{8}$ | ${\href{/padicField/17.2.0.1}{2} }^{4}$ | ${\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.1.2.1a1.1 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 3.1.2.1a1.1 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 3.1.2.1a1.1 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 3.1.2.1a1.1 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
|
\(157\)
| 157.1.2.1a1.1 | $x^{2} + 157$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 157.1.2.1a1.1 | $x^{2} + 157$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 157.1.2.1a1.1 | $x^{2} + 157$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 157.1.2.1a1.1 | $x^{2} + 157$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
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 | 1.157.2t1.a.a | $1$ | $ 157 $ | \(\Q(\sqrt{157}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| *8 | 1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| *8 | 1.471.2t1.a.a | $1$ | $ 3 \cdot 157 $ | \(\Q(\sqrt{-471}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| *16 | 2.471.4t3.b.a | $2$ | $ 3 \cdot 157 $ | 8.0.49213429281.1 | $D_4$ (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 *.