Normalized defining polynomial
\( x^{18} - 7 x^{15} + 1097 x^{12} + 6312 x^{9} + 1101888 x^{6} - 536576 x^{3} + 262144 \)
Invariants
| Degree: | $18$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 9]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-6006399343075824213089699938107=-\,3^{27}\cdot 31^{12}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $51.28$ | magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
| |
| Ramified primes: | $3, 31$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(279=3^{2}\cdot 31\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{279}(1,·)$, $\chi_{279}(67,·)$, $\chi_{279}(5,·)$, $\chi_{279}(32,·)$, $\chi_{279}(211,·)$, $\chi_{279}(149,·)$, $\chi_{279}(25,·)$, $\chi_{279}(218,·)$, $\chi_{279}(94,·)$, $\chi_{279}(160,·)$, $\chi_{279}(98,·)$, $\chi_{279}(125,·)$, $\chi_{279}(242,·)$, $\chi_{279}(118,·)$, $\chi_{279}(56,·)$, $\chi_{279}(187,·)$, $\chi_{279}(253,·)$, $\chi_{279}(191,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{4} a^{8} + \frac{1}{4} a^{5} + \frac{1}{4} a^{2}$, $\frac{1}{8} a^{9} + \frac{1}{8} a^{6} + \frac{1}{8} a^{3}$, $\frac{1}{16} a^{10} + \frac{1}{16} a^{7} + \frac{1}{16} a^{4}$, $\frac{1}{32} a^{11} + \frac{1}{32} a^{8} - \frac{15}{32} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{17728} a^{12} + \frac{993}{17728} a^{9} - \frac{5903}{17728} a^{6} + \frac{325}{1108} a^{3} - \frac{122}{277}$, $\frac{1}{70912} a^{13} - \frac{1223}{70912} a^{10} + \frac{9609}{70912} a^{7} + \frac{2589}{8864} a^{4} - \frac{399}{1108} a$, $\frac{1}{283648} a^{14} - \frac{1223}{283648} a^{11} + \frac{9609}{283648} a^{8} + \frac{11453}{35456} a^{5} + \frac{1817}{4432} a^{2}$, $\frac{1}{21362507763712} a^{15} - \frac{107590791}{21362507763712} a^{12} + \frac{363905363913}{21362507763712} a^{9} + \frac{1205926914949}{2670313470464} a^{6} - \frac{34819017823}{333789183808} a^{3} + \frac{471858849}{5215455997}$, $\frac{1}{85450031054848} a^{16} - \frac{107590791}{85450031054848} a^{13} + \frac{363905363913}{85450031054848} a^{10} + \frac{1205926914949}{10681253881856} a^{7} - \frac{368608201631}{1335156735232} a^{4} - \frac{9959053145}{20861823988} a$, $\frac{1}{341800124219392} a^{17} - \frac{107590791}{341800124219392} a^{14} - \frac{4976721577015}{341800124219392} a^{11} + \frac{538348547333}{42725015527424} a^{8} + \frac{2218257972881}{5340626940928} a^{5} + \frac{10902770843}{83447295952} a^{2}$
Class group and class number
$C_{91}$, which has order $91$ (assuming GRH)
Unit group
| Rank: | $8$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( \frac{1023218599}{341800124219392} a^{17} - \frac{6692027985}{341800124219392} a^{14} + \frac{1118971551711}{341800124219392} a^{11} + \frac{875867309915}{42725015527424} a^{8} + \frac{17658575413967}{5340626940928} a^{5} + \frac{54675803}{10430911994} a^{2} \) (order $18$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| Fundamental units: | Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 4617140.62551 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_3\times C_6$ (as 18T2):
| An abelian group of order 18 |
| The 18 conjugacy class representatives for $C_6 \times C_3$ |
| Character table for $C_6 \times C_3$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 3.3.77841.1, 3.3.77841.2, \(\Q(\zeta_{9})^+\), 3.3.961.1, 6.0.18177663843.2, 6.0.18177663843.1, \(\Q(\zeta_{9})\), 6.0.24935067.1, 9.9.471655843734321.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ | R | ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ | R | ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$ |
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 | Data not computed | ||||||
| $31$ | 31.9.6.1 | $x^{9} + 837 x^{6} + 232562 x^{3} + 21717639$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ |
| 31.9.6.1 | $x^{9} + 837 x^{6} + 232562 x^{3} + 21717639$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ | |