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Properties

Label 18.2.39048002183...2112.1
Degree $18$
Signature $[2, 8]$
Discriminant $2^{12}\cdot 3^{45}\cdot 19^{9}$
Root discriminant $107.86$
Ramified primes $2, 3, 19$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times He_3:C_2$ (as 18T41)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-201675056, 113599728, 136845864, -84735996, -27337428, 20146284, -287778, 144720, -30240, -291299, 40626, 12330, 1545, 900, -450, -51, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 51*x^15 - 450*x^14 + 900*x^13 + 1545*x^12 + 12330*x^11 + 40626*x^10 - 291299*x^9 - 30240*x^8 + 144720*x^7 - 287778*x^6 + 20146284*x^5 - 27337428*x^4 - 84735996*x^3 + 136845864*x^2 + 113599728*x - 201675056)
 
gp: K = bnfinit(x^18 - 51*x^15 - 450*x^14 + 900*x^13 + 1545*x^12 + 12330*x^11 + 40626*x^10 - 291299*x^9 - 30240*x^8 + 144720*x^7 - 287778*x^6 + 20146284*x^5 - 27337428*x^4 - 84735996*x^3 + 136845864*x^2 + 113599728*x - 201675056, 1)
 

Normalized defining polynomial

\( x^{18} - 51 x^{15} - 450 x^{14} + 900 x^{13} + 1545 x^{12} + 12330 x^{11} + 40626 x^{10} - 291299 x^{9} - 30240 x^{8} + 144720 x^{7} - 287778 x^{6} + 20146284 x^{5} - 27337428 x^{4} - 84735996 x^{3} + 136845864 x^{2} + 113599728 x - 201675056 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3904800218300968706245431895711322112=2^{12}\cdot 3^{45}\cdot 19^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $107.86$
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 not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3}$, $\frac{1}{3} a^{10} - \frac{1}{3} a$, $\frac{1}{6} a^{11} - \frac{1}{6} a^{10} - \frac{1}{6} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{6} a^{12} - \frac{1}{6} a^{9} - \frac{1}{2} a^{6} - \frac{1}{6} a^{3} - \frac{1}{3}$, $\frac{1}{6} a^{13} - \frac{1}{6} a^{10} - \frac{1}{2} a^{7} - \frac{1}{6} a^{4} - \frac{1}{3} a$, $\frac{1}{18} a^{14} + \frac{1}{18} a^{13} + \frac{1}{18} a^{12} - \frac{1}{18} a^{11} - \frac{1}{18} a^{10} - \frac{1}{18} a^{9} - \frac{1}{6} a^{8} - \frac{1}{6} a^{7} - \frac{1}{6} a^{6} - \frac{1}{18} a^{5} - \frac{1}{18} a^{4} - \frac{1}{18} a^{3} + \frac{2}{9} a^{2} + \frac{2}{9} a + \frac{2}{9}$, $\frac{1}{36} a^{15} + \frac{1}{36} a^{12} - \frac{1}{6} a^{10} - \frac{5}{36} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{7}{36} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{4}{9} a^{3} - \frac{1}{3} a + \frac{2}{9}$, $\frac{1}{612} a^{16} - \frac{1}{102} a^{15} - \frac{1}{51} a^{14} + \frac{37}{612} a^{13} + \frac{1}{102} a^{12} + \frac{1}{102} a^{11} + \frac{31}{612} a^{10} + \frac{2}{51} a^{9} + \frac{13}{34} a^{8} + \frac{245}{612} a^{7} + \frac{1}{3} a^{6} - \frac{25}{102} a^{5} - \frac{22}{153} a^{4} - \frac{14}{51} a^{3} - \frac{23}{51} a^{2} + \frac{74}{153} a + \frac{22}{51}$, $\frac{1}{568780926994249488743568984140085942217113139441573863240} a^{17} - \frac{1002648590996298926883461395540218267509653165845743}{6463419624934653281176920274319158434285376584563339355} a^{16} + \frac{47575423644209082445578824664808822480278387322137823}{4308946416623102187451280182879438956190251056375559570} a^{15} - \frac{47665003014997629845797611416769001128198535754428909}{2230513439193135249974780329961121342027894664476760248} a^{14} - \frac{516301302708243117441445826308235230000158922215062619}{6319788077713883208261877601556510469079034882684154036} a^{13} + \frac{322559134574681868755367671174506226595008659029185609}{14219523174856237218589224603502148555427828486039346581} a^{12} - \frac{3633179210387868031586627677419763626268685921553629603}{113756185398849897748713796828017188443422627888314772648} a^{11} + \frac{2145297987826816010222724129337310187903754915660113755}{56878092699424948874356898414008594221711313944157386324} a^{10} - \frac{45629996779704528712770242097652044027271762671377027557}{284390463497124744371784492070042971108556569720786931620} a^{9} - \frac{25059813687180689121465204151648806783333578634749096443}{568780926994249488743568984140085942217113139441573863240} a^{8} - \frac{65187908572959696385693495023289825173298794225901295047}{142195231748562372185892246035021485554278284860393465810} a^{7} - \frac{3419556794593176405103101735719397681673405583848001479}{47398410582854124061964082011673828518092761620131155270} a^{6} + \frac{27291688199149089207161035009755730408613201626085539369}{94796821165708248123928164023347657036185523240262310540} a^{5} + \frac{6598739000351006779263221923114011630426849726108383927}{23699205291427062030982041005836914259046380810065577635} a^{4} - \frac{134993188105326938595868263959927331407144231912631241}{28439046349712474437178449207004297110855656972078693162} a^{3} - \frac{39915088252607986035041044447611297690911117522645897039}{142195231748562372185892246035021485554278284860393465810} a^{2} + \frac{65279292973768030489470010173684068587144494253647124}{798849616564957147111754191207985873900439802586480145} a + \frac{764552055184999598645898170854944679897056899374527317}{6463419624934653281176920274319158434285376584563339355}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $9$
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:  \( 1036679926594.8136 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times He_3:C_2$ (as 18T41):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 20 conjugacy class representatives for $C_2\times He_3:C_2$
Character table for $C_2\times He_3:C_2$

Intermediate fields

\(\Q(\sqrt{57}) \), 3.1.243.1, 6.2.1215051273.1, 9.1.2008387814976.4

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 18 siblings: data not computed

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
3Data not computed
$19$19.6.3.2$x^{6} - 361 x^{2} + 27436$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
19.6.3.2$x^{6} - 361 x^{2} + 27436$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
19.6.3.2$x^{6} - 361 x^{2} + 27436$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$