Normalized defining polynomial
\( x^{18} - 9x^{15} + 36x^{12} - 53x^{9} + 27x^{6} + 1 \)
Invariants
| Degree: | $18$ |
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| Signature: | $(0, 9)$ |
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| Discriminant: |
\(-2954312706550833698643\)
\(\medspace = -\,3^{45}\)
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| Root discriminant: | \(15.59\) |
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| Galois root discriminant: | $3^{49/18}\approx 19.898946404249138$ | ||
| Ramified primes: |
\(3\)
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| Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
| $\Aut(K/\Q)$: | $C_9$ |
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| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | \(\Q(\zeta_{9})\) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2071}a^{15}-\frac{393}{2071}a^{12}-\frac{235}{2071}a^{9}-\frac{937}{2071}a^{6}-\frac{519}{2071}a^{3}+\frac{480}{2071}$, $\frac{1}{2071}a^{16}-\frac{393}{2071}a^{13}-\frac{235}{2071}a^{10}-\frac{937}{2071}a^{7}-\frac{519}{2071}a^{4}+\frac{480}{2071}a$, $\frac{1}{2071}a^{17}-\frac{393}{2071}a^{14}-\frac{235}{2071}a^{11}-\frac{937}{2071}a^{8}-\frac{519}{2071}a^{5}+\frac{480}{2071}a^{2}$
| 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: |
\( \frac{350}{2071} a^{15} - \frac{2935}{2071} a^{12} + \frac{10945}{2071} a^{9} - \frac{13158}{2071} a^{6} + \frac{6811}{2071} a^{3} - \frac{1822}{2071} \)
(order $18$)
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| Fundamental units: |
$\frac{350}{2071}a^{16}-\frac{2935}{2071}a^{13}+\frac{10945}{2071}a^{10}-\frac{13158}{2071}a^{7}+\frac{6811}{2071}a^{4}-\frac{3893}{2071}a$, $\frac{6}{109}a^{16}-\frac{69}{109}a^{13}+\frac{334}{109}a^{10}-\frac{717}{109}a^{7}+\frac{483}{109}a^{4}+\frac{46}{109}a$, $\frac{6}{109}a^{17}-\frac{604}{2071}a^{16}-\frac{75}{2071}a^{15}-\frac{69}{109}a^{14}+\frac{5420}{2071}a^{13}+\frac{481}{2071}a^{12}+\frac{334}{109}a^{11}-\frac{21669}{2071}a^{10}-\frac{1014}{2071}a^{9}-\frac{717}{109}a^{8}+\frac{31630}{2071}a^{7}-\frac{2210}{2071}a^{6}+\frac{483}{109}a^{5}-\frac{15813}{2071}a^{4}+\frac{3718}{2071}a^{3}+\frac{46}{109}a^{2}+\frac{20}{2071}a-\frac{793}{2071}$, $\frac{975}{2071}a^{17}+\frac{726}{2071}a^{16}+\frac{451}{2071}a^{15}-\frac{8324}{2071}a^{14}-\frac{5733}{2071}a^{13}-\frac{3279}{2071}a^{12}+\frac{31821}{2071}a^{11}+\frac{19922}{2071}a^{10}+\frac{9991}{2071}a^{9}-\frac{41684}{2071}a^{8}-\frac{17542}{2071}a^{7}-\frac{2174}{2071}a^{6}+\frac{24151}{2071}a^{5}+\frac{4270}{2071}a^{4}-\frac{6259}{2071}a^{3}-\frac{6259}{2071}a^{2}-\frac{1519}{2071}a-\frac{975}{2071}$, $\frac{544}{2071}a^{17}+\frac{1418}{2071}a^{16}+\frac{275}{2071}a^{15}-\frac{4621}{2071}a^{14}-\frac{12601}{2071}a^{13}-\frac{2454}{2071}a^{12}+\frac{17130}{2071}a^{11}+\frac{49905}{2071}a^{10}+\frac{9931}{2071}a^{9}-\frac{18901}{2071}a^{8}-\frac{71569}{2071}a^{7}-\frac{15368}{2071}a^{6}-\frac{680}{2071}a^{5}+\frac{36541}{2071}a^{4}+\frac{10529}{2071}a^{3}+\frac{10529}{2071}a^{2}+\frac{1352}{2071}a-\frac{544}{2071}$, $\frac{653}{2071}a^{17}-\frac{303}{2071}a^{16}-\frac{425}{2071}a^{15}-\frac{6038}{2071}a^{14}+\frac{3103}{2071}a^{13}+\frac{3416}{2071}a^{12}+\frac{24651}{2071}a^{11}-\frac{13706}{2071}a^{10}-\frac{11959}{2071}a^{9}-\frac{38194}{2071}a^{8}+\frac{25036}{2071}a^{7}+\frac{10948}{2071}a^{6}+\frac{19376}{2071}a^{5}-\frac{12565}{2071}a^{4}-\frac{1022}{2071}a^{3}-\frac{1352}{2071}a^{2}-\frac{2541}{2071}a+\frac{1029}{2071}$, $\frac{140}{2071}a^{15}-\frac{1174}{2071}a^{12}+\frac{4378}{2071}a^{9}-\frac{4849}{2071}a^{6}-\frac{175}{2071}a^{3}-a^{2}-a+\frac{928}{2071}$, $\frac{6}{109}a^{17}-\frac{6}{109}a^{16}+\frac{140}{2071}a^{15}-\frac{69}{109}a^{14}+\frac{69}{109}a^{13}-\frac{1174}{2071}a^{12}+\frac{334}{109}a^{11}-\frac{334}{109}a^{10}+\frac{4378}{2071}a^{9}-\frac{717}{109}a^{8}+\frac{717}{109}a^{7}-\frac{4849}{2071}a^{6}+\frac{483}{109}a^{5}-\frac{483}{109}a^{4}-\frac{175}{2071}a^{3}+\frac{46}{109}a^{2}+\frac{63}{109}a+\frac{928}{2071}$
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| Regulator: | \( 23143.3269777 \) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 23143.3269777 \cdot 1}{18\cdot\sqrt{2954312706550833698643}}\cr\approx \mathstrut & 0.361030481189 \end{aligned}\]
Galois group
$S_3\times C_9$ (as 18T16):
| A solvable group of order 54 |
| The 27 conjugacy class representatives for $C_9\times S_3$ |
| Character table for $C_9\times S_3$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{9})\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 27 sibling: | 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 | $18$ | R | $18$ | ${\href{/padicField/7.9.0.1}{9} }^{2}$ | $18$ | ${\href{/padicField/13.9.0.1}{9} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{3}$ | ${\href{/padicField/19.3.0.1}{3} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }^{9}$ | $18$ | $18$ | ${\href{/padicField/31.9.0.1}{9} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }^{9}$ | $18$ | ${\href{/padicField/43.9.0.1}{9} }^{2}$ | $18$ | ${\href{/padicField/53.2.0.1}{2} }^{9}$ | $18$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.1.18.45a2.44 | $x^{18} + 18 x^{17} + 18 x^{16} + 9 x^{14} + 9 x^{13} + 6 x^{12} + 18 x^{10} + 3$ | $18$ | $1$ | $45$ | not computed | not computed |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| *54 | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| *54 | 1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| *54 | 1.9.3t1.a.a | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})^+\) | $C_3$ (as 3T1) | $0$ | $1$ |
| *54 | 1.9.6t1.a.a | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})\) | $C_6$ (as 6T1) | $0$ | $-1$ |
| *54 | 1.9.6t1.a.b | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})\) | $C_6$ (as 6T1) | $0$ | $-1$ |
| *54 | 1.9.3t1.a.b | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})^+\) | $C_3$ (as 3T1) | $0$ | $1$ |
| 1.27.18t1.a.a | $1$ | $ 3^{3}$ | \(\Q(\zeta_{27})\) | $C_{18}$ (as 18T1) | $0$ | $-1$ | |
| 1.27.18t1.a.b | $1$ | $ 3^{3}$ | \(\Q(\zeta_{27})\) | $C_{18}$ (as 18T1) | $0$ | $-1$ | |
| 1.27.18t1.a.c | $1$ | $ 3^{3}$ | \(\Q(\zeta_{27})\) | $C_{18}$ (as 18T1) | $0$ | $-1$ | |
| 1.27.9t1.a.a | $1$ | $ 3^{3}$ | \(\Q(\zeta_{27})^+\) | $C_9$ (as 9T1) | $0$ | $1$ | |
| 1.27.18t1.a.d | $1$ | $ 3^{3}$ | \(\Q(\zeta_{27})\) | $C_{18}$ (as 18T1) | $0$ | $-1$ | |
| 1.27.9t1.a.b | $1$ | $ 3^{3}$ | \(\Q(\zeta_{27})^+\) | $C_9$ (as 9T1) | $0$ | $1$ | |
| 1.27.18t1.a.e | $1$ | $ 3^{3}$ | \(\Q(\zeta_{27})\) | $C_{18}$ (as 18T1) | $0$ | $-1$ | |
| 1.27.9t1.a.c | $1$ | $ 3^{3}$ | \(\Q(\zeta_{27})^+\) | $C_9$ (as 9T1) | $0$ | $1$ | |
| 1.27.9t1.a.d | $1$ | $ 3^{3}$ | \(\Q(\zeta_{27})^+\) | $C_9$ (as 9T1) | $0$ | $1$ | |
| 1.27.9t1.a.e | $1$ | $ 3^{3}$ | \(\Q(\zeta_{27})^+\) | $C_9$ (as 9T1) | $0$ | $1$ | |
| 1.27.9t1.a.f | $1$ | $ 3^{3}$ | \(\Q(\zeta_{27})^+\) | $C_9$ (as 9T1) | $0$ | $1$ | |
| 1.27.18t1.a.f | $1$ | $ 3^{3}$ | \(\Q(\zeta_{27})\) | $C_{18}$ (as 18T1) | $0$ | $-1$ | |
| 2.243.3t2.b.a | $2$ | $ 3^{5}$ | 3.1.243.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
| 2.243.6t5.b.a | $2$ | $ 3^{5}$ | 6.0.177147.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ | |
| 2.243.6t5.b.b | $2$ | $ 3^{5}$ | 6.0.177147.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ | |
| *54 | 2.729.18t16.a.a | $2$ | $ 3^{6}$ | 18.0.2954312706550833698643.2 | $C_9\times S_3$ (as 18T16) | $0$ | $0$ |
| *54 | 2.729.18t16.a.b | $2$ | $ 3^{6}$ | 18.0.2954312706550833698643.2 | $C_9\times S_3$ (as 18T16) | $0$ | $0$ |
| *54 | 2.729.18t16.a.c | $2$ | $ 3^{6}$ | 18.0.2954312706550833698643.2 | $C_9\times S_3$ (as 18T16) | $0$ | $0$ |
| *54 | 2.729.18t16.a.d | $2$ | $ 3^{6}$ | 18.0.2954312706550833698643.2 | $C_9\times S_3$ (as 18T16) | $0$ | $0$ |
| *54 | 2.729.18t16.a.e | $2$ | $ 3^{6}$ | 18.0.2954312706550833698643.2 | $C_9\times S_3$ (as 18T16) | $0$ | $0$ |
| *54 | 2.729.18t16.a.f | $2$ | $ 3^{6}$ | 18.0.2954312706550833698643.2 | $C_9\times S_3$ (as 18T16) | $0$ | $0$ |
Data is given for all irreducible
representations of the Galois group for the Galois closure
of this field. Those marked with * are summands in the
permutation representation coming from this field. Representations
which appear with multiplicity greater than one are indicated
by exponents on the *.