Normalized defining polynomial
\( x^{18} - 6x^{15} + 15x^{12} - 20x^{9} + 22x^{6} - 6x^{3} + 1 \)
Invariants
| Degree: | $18$ |
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| Signature: | $[0, 9]$ |
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| Discriminant: |
\(-1839302601739695422127\)
\(\medspace = -\,3^{18}\cdot 7^{15}\)
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| Root discriminant: | \(15.18\) |
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| Galois root discriminant: | $3^{7/6}7^{5/6}\approx 18.234330849744854$ | ||
| Ramified primes: |
\(3\), \(7\)
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| Discriminant root field: | \(\Q(\sqrt{-7}) \) | ||
| $\Aut(K/\Q)$: | $C_6$ |
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| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | \(\Q(\zeta_{7})\) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{8}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{8}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{58}a^{15}+\frac{4}{29}a^{12}+\frac{11}{58}a^{9}-\frac{11}{58}a^{6}+\frac{13}{58}a^{3}+\frac{1}{29}$, $\frac{1}{58}a^{16}+\frac{4}{29}a^{13}+\frac{11}{58}a^{10}-\frac{11}{58}a^{7}+\frac{13}{58}a^{4}+\frac{1}{29}a$, $\frac{1}{58}a^{17}+\frac{4}{29}a^{14}+\frac{11}{58}a^{11}-\frac{11}{58}a^{8}+\frac{13}{58}a^{5}+\frac{1}{29}a^{2}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ |
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| Narrow class group: | Trivial group, which has order $1$ |
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Unit group
| Rank: | $8$ |
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| Torsion generator: |
\( \frac{19}{58} a^{15} - \frac{109}{58} a^{12} + \frac{267}{58} a^{9} - \frac{177}{29} a^{6} + \frac{196}{29} a^{3} - \frac{49}{58} \)
(order $14$)
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| Fundamental units: |
$\frac{8}{29}a^{17}-\frac{75}{58}a^{14}+\frac{59}{29}a^{11}-\frac{31}{58}a^{8}+\frac{5}{58}a^{5}+\frac{235}{58}a^{2}$, $\frac{7}{29}a^{16}-\frac{91}{58}a^{13}+\frac{241}{58}a^{10}-\frac{164}{29}a^{7}+\frac{327}{58}a^{4}-\frac{73}{29}a$, $\frac{9}{58}a^{17}-\frac{21}{58}a^{16}-\frac{13}{58}a^{15}-\frac{22}{29}a^{14}+\frac{61}{29}a^{13}+\frac{35}{29}a^{12}+\frac{35}{29}a^{11}-\frac{289}{58}a^{10}-\frac{143}{58}a^{9}-\frac{6}{29}a^{8}+\frac{347}{58}a^{7}+\frac{143}{58}a^{6}+\frac{1}{58}a^{5}-\frac{331}{58}a^{4}-\frac{169}{58}a^{3}+\frac{105}{58}a^{2}+\frac{8}{29}a+\frac{16}{29}$, $\frac{13}{29}a^{17}-\frac{9}{58}a^{16}-\frac{21}{58}a^{15}-\frac{169}{58}a^{14}+\frac{22}{29}a^{13}+\frac{61}{29}a^{12}+\frac{230}{29}a^{11}-\frac{35}{29}a^{10}-\frac{289}{58}a^{9}-\frac{663}{58}a^{8}+\frac{6}{29}a^{7}+\frac{347}{58}a^{6}+\frac{715}{58}a^{5}-\frac{1}{58}a^{4}-\frac{331}{58}a^{3}-\frac{325}{58}a^{2}-\frac{105}{58}a-\frac{21}{29}$, $\frac{50}{29}a^{17}-\frac{21}{29}a^{16}+\frac{13}{58}a^{15}-\frac{296}{29}a^{14}+\frac{122}{29}a^{13}-\frac{35}{29}a^{12}+\frac{724}{29}a^{11}-\frac{289}{29}a^{10}+\frac{143}{58}a^{9}-\frac{927}{29}a^{8}+\frac{347}{29}a^{7}-\frac{143}{58}a^{6}+\frac{998}{29}a^{5}-\frac{360}{29}a^{4}+\frac{169}{58}a^{3}-\frac{190}{29}a^{2}+\frac{16}{29}a+\frac{42}{29}$, $\frac{7}{29}a^{17}-\frac{4}{29}a^{15}-\frac{91}{58}a^{14}+\frac{26}{29}a^{12}+\frac{241}{58}a^{11}-\frac{73}{29}a^{9}-\frac{164}{29}a^{8}+\frac{102}{29}a^{6}+\frac{327}{58}a^{5}-\frac{81}{29}a^{3}-\frac{73}{29}a^{2}+\frac{21}{29}$, $\frac{107}{58}a^{17}+\frac{23}{29}a^{16}+\frac{35}{58}a^{15}-\frac{623}{58}a^{14}-\frac{135}{29}a^{13}-\frac{213}{58}a^{12}+\frac{748}{29}a^{11}+\frac{651}{58}a^{10}+\frac{265}{29}a^{9}-\frac{1873}{58}a^{8}-\frac{825}{58}a^{7}-\frac{675}{58}a^{6}+\frac{1000}{29}a^{5}+\frac{444}{29}a^{4}+\frac{358}{29}a^{3}-\frac{125}{29}a^{2}-\frac{169}{58}a-\frac{81}{29}$, $\frac{14}{29}a^{17}+\frac{21}{58}a^{16}-\frac{153}{58}a^{14}-\frac{61}{29}a^{13}+\frac{337}{58}a^{11}+\frac{289}{58}a^{10}-\frac{183}{29}a^{8}-\frac{347}{58}a^{7}+\frac{393}{58}a^{5}+\frac{389}{58}a^{4}+\frac{28}{29}a^{2}-\frac{37}{29}a$
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| Regulator: | \( 13130.466313286553 \) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 13130.466313286553 \cdot 1}{14\cdot\sqrt{1839302601739695422127}}\cr\approx \mathstrut & 0.333767723655804 \end{aligned}\]
Galois group
$C_6\times S_3$ (as 18T6):
| A solvable group of order 36 |
| The 18 conjugacy class representatives for $S_3 \times C_6$ |
| Character table for $S_3 \times C_6$ |
Intermediate fields
| \(\Q(\sqrt{-7}) \), \(\Q(\zeta_{7})^+\), 3.1.1323.1, \(\Q(\zeta_{7})\), 6.0.12252303.1, 9.3.2315685267.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | data not computed |
| Degree 12 sibling: | 12.0.27572864474169.1 |
| Degree 18 sibling: | 18.6.49661170246971776397429.2 |
| Minimal sibling: | 12.0.27572864474169.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.6.0.1}{6} }^{2}{,}\,{\href{/padicField/2.3.0.1}{3} }^{2}$ | R | ${\href{/padicField/5.6.0.1}{6} }^{3}$ | R | ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }^{3}$ | ${\href{/padicField/17.6.0.1}{6} }^{3}$ | ${\href{/padicField/19.6.0.1}{6} }^{3}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{6}{,}\,{\href{/padicField/29.1.0.1}{1} }^{6}$ | ${\href{/padicField/31.6.0.1}{6} }^{3}$ | ${\href{/padicField/37.3.0.1}{3} }^{6}$ | ${\href{/padicField/41.2.0.1}{2} }^{9}$ | ${\href{/padicField/43.3.0.1}{3} }^{6}$ | ${\href{/padicField/47.6.0.1}{6} }^{3}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.6.3.18a2.1 | $x^{18} + 6 x^{16} + 15 x^{14} + 6 x^{13} + 26 x^{12} + 24 x^{11} + 39 x^{10} + 36 x^{9} + 54 x^{8} + 48 x^{7} + 64 x^{6} + 54 x^{5} + 48 x^{4} + 32 x^{3} + 39 x^{2} + 30 x + 17$ | $3$ | $6$ | $18$ | not computed | not computed |
|
\(7\)
| 7.1.6.5a1.1 | $x^{6} + 7$ | $6$ | $1$ | $5$ | $C_6$ | $$[\ ]_{6}$$ |
| 7.1.6.5a1.1 | $x^{6} + 7$ | $6$ | $1$ | $5$ | $C_6$ | $$[\ ]_{6}$$ | |
| 7.1.6.5a1.1 | $x^{6} + 7$ | $6$ | $1$ | $5$ | $C_6$ | $$[\ ]_{6}$$ |