Normalized defining polynomial
\( x^{18} + 570 x^{16} + 123975 x^{14} + 13922250 x^{12} + 890161875 x^{10} + 32905743750 x^{8} + 667386281250 x^{6} + 6305250937500 x^{4} + 16368106640625 x^{2} + 121236328125 \)
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: | \(-21397106810613552737877882027518870016000000000=-\,2^{18}\cdot 3^{27}\cdot 5^{9}\cdot 19^{17}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $374.89$ | 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: | $2, 3, 5, 19$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(3420=2^{2}\cdot 3^{2}\cdot 5\cdot 19\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{3420}(1,·)$, $\chi_{3420}(2761,·)$, $\chi_{3420}(1739,·)$, $\chi_{3420}(301,·)$, $\chi_{3420}(1681,·)$, $\chi_{3420}(2579,·)$, $\chi_{3420}(659,·)$, $\chi_{3420}(2159,·)$, $\chi_{3420}(3359,·)$, $\chi_{3420}(3361,·)$, $\chi_{3420}(3419,·)$, $\chi_{3420}(3241,·)$, $\chi_{3420}(1261,·)$, $\chi_{3420}(3119,·)$, $\chi_{3420}(179,·)$, $\chi_{3420}(841,·)$, $\chi_{3420}(59,·)$, $\chi_{3420}(61,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{5} a^{2}$, $\frac{1}{5} a^{3}$, $\frac{1}{25} a^{4}$, $\frac{1}{25} a^{5}$, $\frac{1}{375} a^{6}$, $\frac{1}{375} a^{7}$, $\frac{1}{1875} a^{8}$, $\frac{1}{20625} a^{9} + \frac{1}{4125} a^{7} + \frac{4}{275} a^{5} + \frac{4}{55} a^{3} + \frac{4}{11} a$, $\frac{1}{103125} a^{10} + \frac{1}{20625} a^{8} + \frac{1}{4125} a^{6} + \frac{4}{275} a^{4} + \frac{4}{55} a^{2}$, $\frac{1}{103125} a^{11} - \frac{4}{11} a$, $\frac{1}{1546875} a^{12} - \frac{1}{11} a^{2}$, $\frac{1}{1546875} a^{13} - \frac{1}{11} a^{3}$, $\frac{1}{97584609375} a^{14} - \frac{21}{197140625} a^{12} + \frac{379}{86741875} a^{10} - \frac{2152}{52045125} a^{8} - \frac{12916}{52045125} a^{6} - \frac{47577}{3469675} a^{4} + \frac{30231}{693935} a^{2} + \frac{461}{1147}$, $\frac{1}{97584609375} a^{15} - \frac{21}{197140625} a^{13} + \frac{379}{86741875} a^{11} + \frac{619}{86741875} a^{9} - \frac{299}{52045125} a^{7} + \frac{2891}{3469675} a^{5} - \frac{58088}{693935} a^{3} - \frac{2958}{12617} a$, $\frac{1}{6198058652839453125} a^{16} - \frac{1622459}{413203910189296875} a^{14} + \frac{12067041259}{82640782037859375} a^{12} + \frac{46947967786}{16528156407571875} a^{10} - \frac{53808214811}{661126256302875} a^{8} + \frac{127115507127}{220375418767625} a^{6} + \frac{168551946158}{8815016750705} a^{4} - \frac{144596562973}{8815016750705} a^{2} - \frac{6842744006}{14570275621}$, $\frac{1}{6198058652839453125} a^{17} - \frac{1622459}{413203910189296875} a^{15} + \frac{12067041259}{82640782037859375} a^{13} + \frac{46947967786}{16528156407571875} a^{11} + \frac{17168329869}{1101877093838125} a^{9} + \frac{701892585043}{661126256302875} a^{7} + \frac{361940635297}{44075083753525} a^{5} - \frac{625415658466}{8815016750705} a^{3} + \frac{3753820082}{14570275621} a$
Class group and class number
$C_{2}\times C_{2}\times C_{682763844}$, which has order $2731055376$ (assuming GRH)
Unit group
| Rank: | $8$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -1 \) (order $2$) | 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: | \( 15010229.973756868 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A cyclic group of order 18 |
| The 18 conjugacy class representatives for $C_{18}$ |
| Character table for $C_{18}$ |
Intermediate fields
| \(\Q(\sqrt{-285}) \), 3.3.361.1, 6.0.534837384000.9, 9.9.9025761726072081.1 |
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 | R | R | R | ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/11.1.0.1}{1} }^{18}$ | ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ | R | $18$ | ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/31.1.0.1}{1} }^{18}$ | ${\href{/LocalNumberField/37.1.0.1}{1} }^{18}$ | ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ | $18$ | $18$ | $18$ |
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 |
|---|---|---|---|---|---|---|---|
| 2 | Data not computed | ||||||
| 3 | Data not computed | ||||||
| 5 | Data not computed | ||||||
| 19 | Data not computed | ||||||