Normalized defining polynomial
\( x^{8} - 3x^{7} - 26x^{6} + 182x^{5} - 349x^{4} - 134x^{3} + 1152x^{2} - 785x - 989 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[4, 2]$ |
| |
| Discriminant: |
\(498053903884848\)
\(\medspace = 2^{4}\cdot 3^{5}\cdot 71^{6}\)
|
| |
| Root discriminant: | \(68.73\) |
| |
| Galois root discriminant: | $2^{3/2}3^{3/4}71^{3/4}\approx 157.6992747634948$ | ||
| Ramified primes: |
\(2\), \(3\), \(71\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{3}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{24761110}a^{7}-\frac{3574411}{24761110}a^{6}+\frac{4788647}{24761110}a^{5}+\frac{4801398}{12380555}a^{4}-\frac{1041291}{12380555}a^{3}+\frac{3359247}{24761110}a^{2}-\frac{941209}{24761110}a-\frac{98961}{1076570}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{3}$, which has order $3$ |
| |
| Narrow class group: | $C_{6}$, which has order $6$ |
|
Unit group
| Rank: | $5$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{24812}{12380555}a^{7}-\frac{359979}{24761110}a^{6}-\frac{276971}{12380555}a^{5}+\frac{13967309}{24761110}a^{4}-\frac{55078803}{24761110}a^{3}+\frac{94144493}{24761110}a^{2}-\frac{28312088}{12380555}a-\frac{1143179}{1076570}$, $\frac{301647}{24761110}a^{7}-\frac{100261}{12380555}a^{6}-\frac{7872461}{24761110}a^{5}+\frac{38054437}{24761110}a^{4}-\frac{27632409}{24761110}a^{3}-\frac{51774858}{12380555}a^{2}+\frac{220866027}{24761110}a+\frac{3959234}{538285}$, $\frac{211107}{24761110}a^{7}-\frac{368141}{12380555}a^{6}-\frac{4935741}{24761110}a^{5}+\frac{37760027}{24761110}a^{4}-\frac{100794109}{24761110}a^{3}+\frac{56895512}{12380555}a^{2}-\frac{12661723}{24761110}a-\frac{2143486}{538285}$, $\frac{19556}{12380555}a^{7}-\frac{567986}{12380555}a^{6}+\frac{262712}{12380555}a^{5}+\frac{16400891}{12380555}a^{4}-\frac{69230972}{12380555}a^{3}+\frac{76492832}{12380555}a^{2}+\frac{40743746}{12380555}a-\frac{1223311}{538285}$, $\frac{36138}{12380555}a^{7}-\frac{5734403}{12380555}a^{6}-\frac{3272504}{12380555}a^{5}+\frac{172212968}{12380555}a^{4}-\frac{457035006}{12380555}a^{3}-\frac{366290339}{12380555}a^{2}+\frac{1654968113}{12380555}a+\frac{43714007}{538285}$
|
| |
| Regulator: | \( 20825.5219384 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 20825.5219384 \cdot 3}{2\cdot\sqrt{498053903884848}}\cr\approx \mathstrut & 0.884155593613 \end{aligned}\]
Galois group
$C_2\wr D_4$ (as 8T35):
| A solvable group of order 128 |
| The 20 conjugacy class representatives for $C_2 \wr C_2\wr C_2$ |
| Character table for $C_2 \wr C_2\wr C_2$ |
Intermediate fields
| \(\Q(\sqrt{213}) \), 4.2.3221199.1 |
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: | 8.2.336712498401024.4 |
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 | ${\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.8.0.1}{8} }$ | ${\href{/padicField/11.2.0.1}{2} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{3}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.8.0.1}{8} }$ | ${\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{6}$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.2.4a2.2 | $x^{4} + 4 x^{3} + 5 x^{2} + 4 x + 7$ | $2$ | $2$ | $4$ | $D_{4}$ | $$[2, 2]^{2}$$ |
| 2.4.1.0a1.1 | $x^{4} + x + 1$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ | |
|
\(3\)
| 3.2.2.2a1.2 | $x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ |
| 3.1.4.3a1.1 | $x^{4} + 3$ | $4$ | $1$ | $3$ | $D_{4}$ | $$[\ ]_{4}^{2}$$ | |
|
\(71\)
| 71.2.4.6a1.2 | $x^{8} + 276 x^{7} + 28594 x^{6} + 1319832 x^{5} + 23067339 x^{4} + 9238824 x^{3} + 1401106 x^{2} + 94668 x + 2472$ | $4$ | $2$ | $6$ | $D_4$ | $$[\ ]_{4}^{2}$$ |