Normalized defining polynomial
\( x^{18} + 342 x^{16} + 36423 x^{14} + 1398894 x^{12} + 25743195 x^{10} + 255741786 x^{8} + 1406615562 x^{6} + 4024770228 x^{4} + 4661278569 x^{2} + 5640625 \)
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: | \(-74461665313712352487171933214411650949175312384=-\,2^{18}\cdot 3^{44}\cdot 19^{16}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $401.79$ | 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: | $2, 3, 19$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(2052=2^{2}\cdot 3^{3}\cdot 19\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{2052}(1,·)$, $\chi_{2052}(1027,·)$, $\chi_{2052}(577,·)$, $\chi_{2052}(139,·)$, $\chi_{2052}(1165,·)$, $\chi_{2052}(727,·)$, $\chi_{2052}(1603,·)$, $\chi_{2052}(853,·)$, $\chi_{2052}(1879,·)$, $\chi_{2052}(1753,·)$, $\chi_{2052}(871,·)$, $\chi_{2052}(1897,·)$, $\chi_{2052}(427,·)$, $\chi_{2052}(1453,·)$, $\chi_{2052}(175,·)$, $\chi_{2052}(1201,·)$, $\chi_{2052}(505,·)$, $\chi_{2052}(1531,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{5} a^{5} - \frac{1}{5} a$, $\frac{1}{5} a^{6} - \frac{1}{5} a^{2}$, $\frac{1}{5} a^{7} - \frac{1}{5} a^{3}$, $\frac{1}{5} a^{8} - \frac{1}{5} a^{4}$, $\frac{1}{475} a^{9} + \frac{2}{25} a^{5} - \frac{6}{25} a$, $\frac{1}{475} a^{10} + \frac{2}{25} a^{6} - \frac{6}{25} a^{2}$, $\frac{1}{475} a^{11} + \frac{2}{25} a^{7} - \frac{6}{25} a^{3}$, $\frac{1}{475} a^{12} + \frac{2}{25} a^{8} - \frac{6}{25} a^{4}$, $\frac{1}{2375} a^{13} + \frac{2}{2375} a^{9} - \frac{3}{125} a^{5} + \frac{16}{125} a$, $\frac{1}{2434375} a^{14} - \frac{242}{486875} a^{12} + \frac{1442}{2434375} a^{10} - \frac{1104}{25625} a^{8} + \frac{6952}{128125} a^{6} - \frac{10828}{25625} a^{4} + \frac{39176}{128125} a^{2} + \frac{19}{41}$, $\frac{1}{2434375} a^{15} - \frac{37}{486875} a^{13} + \frac{1442}{2434375} a^{11} - \frac{66}{486875} a^{9} + \frac{6952}{128125} a^{7} - \frac{1193}{25625} a^{5} + \frac{39176}{128125} a^{3} - \frac{44}{5125} a$, $\frac{1}{4936531563466215265625} a^{16} - \frac{369988864477674}{4936531563466215265625} a^{14} - \frac{1870895058258375368}{4936531563466215265625} a^{12} + \frac{3615324675649603282}{4936531563466215265625} a^{10} + \frac{2227795693850919232}{259817450708748171875} a^{8} - \frac{1002890274659725243}{259817450708748171875} a^{6} + \frac{98871727233305090261}{259817450708748171875} a^{4} + \frac{6229314821107790836}{259817450708748171875} a^{2} + \frac{385906977804629}{16628316845359883}$, $\frac{1}{4936531563466215265625} a^{17} - \frac{369988864477674}{4936531563466215265625} a^{15} + \frac{207644547411610007}{4936531563466215265625} a^{13} + \frac{3615324675649603282}{4936531563466215265625} a^{11} + \frac{4914405281107728658}{4936531563466215265625} a^{9} - \frac{1002890274659725243}{259817450708748171875} a^{7} + \frac{9494524189495719136}{259817450708748171875} a^{5} + \frac{6229314821107790836}{259817450708748171875} a^{3} + \frac{231149857524537338}{2078539605669985375} a$
Class group and class number
$C_{161970201}$, which has order $161970201$ (assuming GRH)
Unit group
| Rank: | $8$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -\frac{106990956}{850321697984375} a^{17} - \frac{35793065466}{850321697984375} a^{15} - \frac{3630434636592}{850321697984375} a^{13} - \frac{122734353245112}{850321697984375} a^{11} - \frac{1852782228916123}{850321697984375} a^{9} - \frac{736191985340187}{44753773578125} a^{7} - \frac{67227115544451}{1091555453125} a^{5} - \frac{4339543074810276}{44753773578125} a^{3} - \frac{10328848430643}{358030188625} a \) (order $4$) | 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: | \( 5772307958.489205 \) (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
| \(\Q(\sqrt{-1}) \), 3.3.29241.2, 6.0.54722309184.1, 9.9.532962204162830310969.6 |
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 | ${\href{/LocalNumberField/5.1.0.1}{1} }^{18}$ | $18$ | $18$ | ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ | R | $18$ | ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ | $18$ | ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/41.1.0.1}{1} }^{18}$ | $18$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ | ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
| 2 | Data not computed | ||||||
| 3 | Data not computed | ||||||
| 19 | Data not computed | ||||||