Normalized defining polynomial
\( x^{8} + 8x^{6} - 16x^{4} - 1628x^{2} - 1369 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[2, 3]$ |
| |
| Discriminant: |
\(-897187840000\)
\(\medspace = -\,2^{20}\cdot 5^{4}\cdot 37^{2}\)
|
| |
| Root discriminant: | \(31.20\) |
| |
| Galois root discriminant: | $2^{51/16}5^{1/2}37^{1/2}\approx 123.91360025060992$ | ||
| Ramified primes: |
\(2\), \(5\), \(37\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-1}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{10}a^{4}+\frac{2}{5}a^{2}-\frac{1}{10}$, $\frac{1}{10}a^{5}+\frac{2}{5}a^{3}-\frac{1}{10}a$, $\frac{1}{31450}a^{6}+\frac{263}{15725}a^{4}-\frac{7453}{31450}a^{2}-\frac{173}{425}$, $\frac{1}{31450}a^{7}+\frac{263}{15725}a^{5}-\frac{7453}{31450}a^{3}-\frac{173}{425}a$
| 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: | $C_{4}$, which has order $4$ |
|
Unit group
| Rank: | $4$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{3}{3145}a^{6}+\frac{11}{6290}a^{4}-\frac{344}{3145}a^{2}-\frac{121}{170}$, $\frac{13}{31450}a^{6}+\frac{274}{15725}a^{4}+\frac{3751}{31450}a^{2}-\frac{39}{425}$, $\frac{29}{31450}a^{7}-\frac{29}{6290}a^{6}-\frac{471}{31450}a^{5}-\frac{79}{3145}a^{4}+\frac{4013}{31450}a^{3}+\frac{6051}{6290}a^{2}-\frac{259}{850}a-\frac{32}{85}$, $\frac{81}{31450}a^{7}+\frac{176}{15725}a^{6}+\frac{1721}{31450}a^{5}+\frac{5887}{31450}a^{4}+\frac{19017}{31450}a^{3}+\frac{28042}{15725}a^{2}+\frac{1129}{850}a+\frac{1203}{850}$
|
| |
| Regulator: | \( 218.183040545 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{3}\cdot 218.183040545 \cdot 2}{2\cdot\sqrt{897187840000}}\cr\approx \mathstrut & 0.228548774395 \end{aligned}\]
Galois group
| A solvable group of order 64 |
| The 16 conjugacy class representatives for $(C_4^2 : C_2):C_2$ |
| Character table for $(C_4^2 : C_2):C_2$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 4.2.1600.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 8 siblings: | data not computed |
| Degree 16 siblings: | data not computed |
| Degree 32 siblings: | data not computed |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.8.0.1}{8} }$ | R | ${\href{/padicField/7.8.0.1}{8} }$ | ${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{4}$ | ${\href{/padicField/17.2.0.1}{2} }^{4}$ | ${\href{/padicField/19.2.0.1}{2} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }$ | ${\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }$ | ${\href{/padicField/47.8.0.1}{8} }$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.4.20a2.19 | $x^{8} + 8 x^{7} + 22 x^{6} + 40 x^{5} + 51 x^{4} + 48 x^{3} + 42 x^{2} + 24 x + 15$ | $4$ | $2$ | $20$ | $(C_4^2 : C_2):C_2$ | $$[2, 2, 3, \frac{7}{2}, \frac{7}{2}]^{2}$$ |
|
\(5\)
| 5.2.2.2a1.2 | $x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ |
| 5.2.2.2a1.2 | $x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
|
\(37\)
| 37.2.1.0a1.1 | $x^{2} + 33 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 37.2.1.0a1.1 | $x^{2} + 33 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 37.2.2.2a1.2 | $x^{4} + 66 x^{3} + 1093 x^{2} + 132 x + 41$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ |