Normalized defining polynomial
\( x^{8} - 2x^{7} + 2x^{6} - 90x^{5} + 2754x^{4} - 810x^{3} + 162x^{2} - 1458x + 6561 \)
Invariants
| Degree: | $8$ |
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| Signature: | $[0, 4]$ |
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| Discriminant: |
\(524698762940416\)
\(\medspace = 2^{12}\cdot 71^{6}\)
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| Root discriminant: | \(69.18\) |
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| Galois root discriminant: | $2^{2}71^{3/4}\approx 97.83714091452357$ | ||
| Ramified primes: |
\(2\), \(71\)
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| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$: | $C_2^2$ |
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| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | \(\Q(i, \sqrt{71})\) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{18}a^{4}-\frac{1}{9}a^{3}+\frac{1}{9}a^{2}-\frac{1}{2}$, $\frac{1}{54}a^{5}+\frac{1}{54}a^{4}-\frac{2}{27}a^{3}+\frac{4}{9}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{17172}a^{6}+\frac{13}{4293}a^{5}-\frac{61}{17172}a^{4}+\frac{52}{477}a^{3}+\frac{469}{1908}a^{2}+\frac{13}{53}a-\frac{97}{212}$, $\frac{1}{6181920}a^{7}+\frac{5}{1236384}a^{6}+\frac{39557}{6181920}a^{5}-\frac{139}{686880}a^{4}+\frac{8497}{76320}a^{3}-\frac{16831}{76320}a^{2}-\frac{6215}{15264}a-\frac{27}{8480}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$, $3$ |
Class group and class number
| Ideal class group: | $C_{70}$, which has order $70$ |
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| Narrow class group: | $C_{70}$, which has order $70$ |
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Unit group
| Rank: | $3$ |
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| Torsion generator: |
\( -\frac{187}{6181920} a^{7} - \frac{215}{1236384} a^{6} + \frac{2281}{6181920} a^{5} + \frac{59}{25440} a^{4} - \frac{531}{8480} a^{3} - \frac{48523}{76320} a^{2} + \frac{1421}{15264} a + \frac{169}{8480} \)
(order $4$)
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| Fundamental units: |
$\frac{137}{3090960}a^{7}-\frac{107}{618192}a^{6}+\frac{4549}{3090960}a^{5}+\frac{479}{114480}a^{4}+\frac{20747}{114480}a^{3}-\frac{10607}{38160}a^{2}+\frac{6665}{7632}a-\frac{1299}{4240}$, $\frac{8557}{6181920}a^{7}-\frac{9919}{1236384}a^{6}+\frac{120929}{6181920}a^{5}-\frac{102943}{686880}a^{4}+\frac{982807}{228960}a^{3}-\frac{410449}{25440}a^{2}+\frac{335653}{15264}a-\frac{108719}{8480}$, $\frac{39590871413}{6181920}a^{7}+\frac{42331315273}{1236384}a^{6}-\frac{1327336668359}{6181920}a^{5}-\frac{229239154567}{686880}a^{4}+\frac{1018873795781}{76320}a^{3}+\frac{10381114384117}{76320}a^{2}-\frac{6004273154755}{15264}a-\frac{1847068737111}{8480}$
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| Regulator: | \( 5025.96383193 \) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 5025.96383193 \cdot 70}{4\cdot\sqrt{524698762940416}}\cr\approx \mathstrut & 5.98441717890 \end{aligned}\]
Galois group
$C_2\times D_4$ (as 8T9):
| A solvable group of order 16 |
| The 10 conjugacy class representatives for $D_4\times C_2$ |
| Character table for $D_4\times C_2$ |
Intermediate fields
| \(\Q(\sqrt{71}) \), \(\Q(\sqrt{-71}) \), \(\Q(\sqrt{-1}) \), 4.0.11453152.2, 4.0.2863288.1, \(\Q(i, \sqrt{71})\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | deg 16 |
| Degree 8 siblings: | 8.0.2098795051761664.11, 8.4.8395180207046656.2, 8.0.8395180207046656.13 |
| 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 | ${\href{/padicField/3.2.0.1}{2} }^{4}$ | ${\href{/padicField/5.2.0.1}{2} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{4}$ | ${\href{/padicField/7.2.0.1}{2} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.2.2a1.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $$[2]$$ |
| 2.1.2.2a1.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $$[2]$$ | |
| 2.1.4.8b1.1 | $x^{4} + 2 x^{2} + 4 x + 2$ | $4$ | $1$ | $8$ | $C_2^2$ | $$[2, 3]$$ | |
|
\(71\)
| 71.2.4.6a1.2 | $x^{8} + 276 x^{7} + 28594 x^{6} + 1319832 x^{5} + 23067339 x^{4} + 9238824 x^{3} + 1401106 x^{2} + 94668 x + 2472$ | $4$ | $2$ | $6$ | $D_4$ | $$[\ ]_{4}^{2}$$ |