Normalized defining polynomial
\( x^{18} + 98x^{12} + 268x^{6} + 216 \)
Invariants
| Degree: | $18$ |
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| Signature: | $[0, 9]$ |
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| Discriminant: |
\(-21936950640377856000000000000\)
\(\medspace = -\,2^{33}\cdot 3^{21}\cdot 5^{12}\)
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| Root discriminant: | \(37.54\) |
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| Galois root discriminant: | $2^{11/6}3^{7/6}5^{2/3}\approx 37.54134188892015$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\)
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| Discriminant root field: | \(\Q(\sqrt{-6}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
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| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | \(\Q(\sqrt{-6}) \) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{4}a^{6}-\frac{1}{2}$, $\frac{1}{4}a^{7}-\frac{1}{2}a$, $\frac{1}{12}a^{8}-\frac{1}{6}a^{2}$, $\frac{1}{12}a^{9}-\frac{1}{6}a^{3}$, $\frac{1}{72}a^{10}+\frac{11}{36}a^{4}$, $\frac{1}{72}a^{11}+\frac{11}{36}a^{5}$, $\frac{1}{144}a^{12}+\frac{1}{36}a^{6}+\frac{1}{4}$, $\frac{1}{144}a^{13}+\frac{1}{36}a^{7}+\frac{1}{4}a$, $\frac{1}{288}a^{14}+\frac{1}{72}a^{8}-\frac{3}{8}a^{2}$, $\frac{1}{864}a^{15}+\frac{1}{432}a^{13}+\frac{1}{216}a^{11}-\frac{5}{216}a^{9}+\frac{5}{54}a^{7}+\frac{11}{108}a^{5}-\frac{5}{72}a^{3}-\frac{1}{12}a$, $\frac{1}{864}a^{16}-\frac{1}{864}a^{14}-\frac{1}{432}a^{12}+\frac{1}{216}a^{10}-\frac{1}{216}a^{8}+\frac{2}{27}a^{6}-\frac{11}{24}a^{4}+\frac{11}{24}a^{2}-\frac{1}{4}$, $\frac{1}{864}a^{17}-\frac{1}{216}a^{11}-\frac{1}{36}a^{9}-\frac{1}{12}a^{7}+\frac{73}{216}a^{5}+\frac{7}{18}a^{3}+\frac{1}{6}a$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{2}\times C_{6}$, which has order $12$ (assuming GRH) |
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| Narrow class group: | $C_{2}\times C_{6}$, which has order $12$ (assuming GRH) |
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Unit group
| Rank: | $8$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
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| Fundamental units: |
$\frac{1}{72}a^{10}+\frac{47}{36}a^{4}$, $\frac{13}{864}a^{16}-\frac{7}{864}a^{14}+\frac{5}{432}a^{12}+\frac{313}{216}a^{10}-\frac{169}{216}a^{8}+\frac{61}{54}a^{6}+\frac{115}{72}a^{4}-\frac{31}{24}a^{2}+\frac{5}{4}$, $\frac{5}{288}a^{16}+\frac{121}{72}a^{10}+\frac{185}{72}a^{4}+1$, $\frac{1}{432}a^{16}-\frac{11}{864}a^{14}-\frac{5}{432}a^{12}+\frac{25}{108}a^{10}-\frac{263}{216}a^{8}-\frac{61}{54}a^{6}+\frac{35}{36}a^{4}-\frac{5}{8}a^{2}-\frac{5}{4}$, $\frac{1}{288}a^{17}+\frac{5}{288}a^{16}+\frac{13}{864}a^{15}-\frac{1}{216}a^{13}+\frac{73}{216}a^{11}+\frac{121}{72}a^{10}+\frac{313}{216}a^{9}-\frac{47}{108}a^{7}+\frac{163}{216}a^{5}+\frac{185}{72}a^{4}+\frac{115}{72}a^{3}+\frac{2}{3}a+1$, $\frac{5}{288}a^{17}-\frac{5}{288}a^{16}+\frac{5}{3}a^{11}-\frac{121}{72}a^{10}+\frac{91}{72}a^{5}-\frac{185}{72}a^{4}-1$, $\frac{1}{48}a^{17}+\frac{1}{432}a^{16}-\frac{13}{432}a^{15}-\frac{11}{864}a^{14}+\frac{19}{432}a^{13}-\frac{5}{432}a^{12}+\frac{109}{54}a^{11}+\frac{11}{54}a^{10}-\frac{313}{108}a^{9}-\frac{263}{216}a^{8}+\frac{115}{27}a^{7}-\frac{61}{54}a^{6}+\frac{359}{108}a^{5}-\frac{59}{36}a^{4}-\frac{115}{36}a^{3}-\frac{5}{8}a^{2}+\frac{77}{12}a-\frac{13}{4}$, $\frac{25}{432}a^{17}+\frac{47}{432}a^{16}+\frac{23}{288}a^{15}+\frac{7}{432}a^{14}+\frac{7}{144}a^{13}+\frac{17}{432}a^{12}+\frac{152}{27}a^{11}+\frac{284}{27}a^{10}+\frac{553}{72}a^{9}+\frac{169}{108}a^{8}+\frac{43}{9}a^{7}+\frac{431}{108}a^{6}+\frac{1279}{108}a^{5}+\frac{63}{4}a^{4}+\frac{527}{72}a^{3}-\frac{5}{12}a^{2}+\frac{55}{12}a-\frac{1}{4}$
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| Regulator: | \( 2451084.753515054 \) (assuming GRH) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 2451084.753515054 \cdot 12}{2\cdot\sqrt{21936950640377856000000000000}}\cr\approx \mathstrut & 1.51544529993536 \end{aligned}\] (assuming GRH)
Galois group
| A solvable group of order 36 |
| The 12 conjugacy class representatives for $C_6:S_3$ |
| Character table for $C_6:S_3$ |
Intermediate fields
| \(\Q(\sqrt{-6}) \), 3.1.2700.1, 3.1.300.1, 3.1.675.1, 3.1.108.1, 6.0.2799360000.8, 6.0.699840000.2, 6.0.34560000.1, 6.0.4478976.2, 9.1.59049000000.1 |
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 18 sibling: | 18.2.7312316880125952000000000000.1 |
| Minimal sibling: | 18.2.7312316880125952000000000000.1 |
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 | R | ${\href{/padicField/7.3.0.1}{3} }^{6}$ | ${\href{/padicField/11.2.0.1}{2} }^{8}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }^{3}$ | ${\href{/padicField/17.2.0.1}{2} }^{9}$ | ${\href{/padicField/19.6.0.1}{6} }^{3}$ | ${\href{/padicField/23.2.0.1}{2} }^{9}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }^{6}$ | ${\href{/padicField/37.6.0.1}{6} }^{3}$ | ${\href{/padicField/41.2.0.1}{2} }^{9}$ | ${\href{/padicField/43.6.0.1}{6} }^{3}$ | ${\href{/padicField/47.2.0.1}{2} }^{9}$ | ${\href{/padicField/53.2.0.1}{2} }^{8}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.6.11a1.6 | $x^{6} + 4 x^{3} + 10$ | $6$ | $1$ | $11$ | $D_{6}$ | $$[3]_{3}^{2}$$ |
| 2.2.6.22a1.9 | $x^{12} + 6 x^{11} + 21 x^{10} + 50 x^{9} + 90 x^{8} + 126 x^{7} + 145 x^{6} + 138 x^{5} + 114 x^{4} + 78 x^{3} + 45 x^{2} + 18 x + 7$ | $6$ | $2$ | $22$ | $D_6$ | $$[3]_{3}^{2}$$ | |
|
\(3\)
| 3.1.6.7a2.2 | $x^{6} + 3 x^{2} + 6$ | $6$ | $1$ | $7$ | $D_{6}$ | $$[\frac{3}{2}]_{2}^{2}$$ |
| 3.2.6.14a1.3 | $x^{12} + 12 x^{11} + 72 x^{10} + 280 x^{9} + 780 x^{8} + 1632 x^{7} + 2624 x^{6} + 3264 x^{5} + 3123 x^{4} + 2252 x^{3} + 1176 x^{2} + 408 x + 79$ | $6$ | $2$ | $14$ | $D_6$ | $$[\frac{3}{2}]_{2}^{2}$$ | |
|
\(5\)
| 5.1.3.2a1.1 | $x^{3} + 5$ | $3$ | $1$ | $2$ | $S_3$ | $$[\ ]_{3}^{2}$$ |
| 5.1.3.2a1.1 | $x^{3} + 5$ | $3$ | $1$ | $2$ | $S_3$ | $$[\ ]_{3}^{2}$$ | |
| 5.2.3.4a1.2 | $x^{6} + 12 x^{5} + 54 x^{4} + 112 x^{3} + 108 x^{2} + 48 x + 13$ | $3$ | $2$ | $4$ | $S_3$ | $$[\ ]_{3}^{2}$$ | |
| 5.2.3.4a1.2 | $x^{6} + 12 x^{5} + 54 x^{4} + 112 x^{3} + 108 x^{2} + 48 x + 13$ | $3$ | $2$ | $4$ | $S_3$ | $$[\ ]_{3}^{2}$$ |