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Properties

Label 18.2.65619340392...4864.1
Degree $18$
Signature $[2, 8]$
Discriminant $2^{16}\cdot 3^{33}\cdot 23^{9}$
Root discriminant $66.55$
Ramified primes $2, 3, 23$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_2\times D_9:C_3$ (as 18T45)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-5859375, -10546875, -4921875, 5062500, -3262500, -2328750, -1083750, 423450, -476595, -196803, -69381, 17208, -2232, -1944, -108, 0, 21, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 21*x^16 - 108*x^14 - 1944*x^13 - 2232*x^12 + 17208*x^11 - 69381*x^10 - 196803*x^9 - 476595*x^8 + 423450*x^7 - 1083750*x^6 - 2328750*x^5 - 3262500*x^4 + 5062500*x^3 - 4921875*x^2 - 10546875*x - 5859375)
 
gp: K = bnfinit(x^18 - 9*x^17 + 21*x^16 - 108*x^14 - 1944*x^13 - 2232*x^12 + 17208*x^11 - 69381*x^10 - 196803*x^9 - 476595*x^8 + 423450*x^7 - 1083750*x^6 - 2328750*x^5 - 3262500*x^4 + 5062500*x^3 - 4921875*x^2 - 10546875*x - 5859375, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 21 x^{16} - 108 x^{14} - 1944 x^{13} - 2232 x^{12} + 17208 x^{11} - 69381 x^{10} - 196803 x^{9} - 476595 x^{8} + 423450 x^{7} - 1083750 x^{6} - 2328750 x^{5} - 3262500 x^{4} + 5062500 x^{3} - 4921875 x^{2} - 10546875 x - 5859375 \)

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:  \(656193403924348462564108543524864=2^{16}\cdot 3^{33}\cdot 23^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $66.55$
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:  $2, 3, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5} a^{10} + \frac{1}{5} a^{9} + \frac{1}{5} a^{8} + \frac{2}{5} a^{6} + \frac{1}{5} a^{5} - \frac{2}{5} a^{4} - \frac{2}{5} a^{3} - \frac{1}{5} a^{2} + \frac{2}{5} a$, $\frac{1}{25} a^{11} + \frac{1}{25} a^{10} + \frac{6}{25} a^{9} + \frac{2}{5} a^{8} - \frac{8}{25} a^{7} + \frac{1}{25} a^{6} + \frac{3}{25} a^{5} - \frac{12}{25} a^{4} - \frac{1}{25} a^{3} + \frac{12}{25} a^{2}$, $\frac{1}{125} a^{12} + \frac{1}{125} a^{11} + \frac{6}{125} a^{10} + \frac{7}{25} a^{9} - \frac{33}{125} a^{8} - \frac{24}{125} a^{7} - \frac{22}{125} a^{6} - \frac{37}{125} a^{5} + \frac{49}{125} a^{4} - \frac{13}{125} a^{3} + \frac{2}{5} a^{2} + \frac{1}{5} a$, $\frac{1}{625} a^{13} + \frac{1}{625} a^{12} + \frac{6}{625} a^{11} + \frac{7}{125} a^{10} + \frac{92}{625} a^{9} - \frac{24}{625} a^{8} + \frac{228}{625} a^{7} + \frac{88}{625} a^{6} + \frac{174}{625} a^{5} + \frac{237}{625} a^{4} + \frac{7}{25} a^{3} - \frac{4}{25} a^{2} + \frac{2}{5} a$, $\frac{1}{3125} a^{14} + \frac{1}{3125} a^{13} + \frac{6}{3125} a^{12} + \frac{7}{625} a^{11} + \frac{92}{3125} a^{10} + \frac{1226}{3125} a^{9} + \frac{1478}{3125} a^{8} + \frac{1338}{3125} a^{7} + \frac{799}{3125} a^{6} - \frac{388}{3125} a^{5} - \frac{18}{125} a^{4} + \frac{21}{125} a^{3} + \frac{12}{25} a^{2} + \frac{2}{5} a$, $\frac{1}{15625} a^{15} + \frac{1}{15625} a^{14} + \frac{6}{15625} a^{13} + \frac{7}{3125} a^{12} + \frac{92}{15625} a^{11} + \frac{1226}{15625} a^{10} - \frac{4772}{15625} a^{9} + \frac{4463}{15625} a^{8} + \frac{799}{15625} a^{7} + \frac{2737}{15625} a^{6} + \frac{107}{625} a^{5} + \frac{21}{625} a^{4} + \frac{37}{125} a^{3} - \frac{3}{25} a^{2}$, $\frac{1}{5546875} a^{16} + \frac{81}{5546875} a^{15} - \frac{489}{5546875} a^{14} - \frac{262}{1109375} a^{13} + \frac{21942}{5546875} a^{12} - \frac{92164}{5546875} a^{11} + \frac{229783}{5546875} a^{10} + \frac{662128}{5546875} a^{9} - \frac{198886}{5546875} a^{8} + \frac{2639182}{5546875} a^{7} + \frac{219067}{1109375} a^{6} + \frac{14406}{44375} a^{5} + \frac{3933}{44375} a^{4} - \frac{183}{8875} a^{3} - \frac{81}{1775} a^{2} + \frac{156}{355} a - \frac{28}{71}$, $\frac{1}{1958996717612895185875074970771435166172496484375} a^{17} - \frac{135319506809027065621126971210348407898964}{1958996717612895185875074970771435166172496484375} a^{16} - \frac{635051427892854323818921753170158628408229}{27591503064970354730634858743259650227781640625} a^{15} - \frac{21856401715550890127286310872666594919605286}{391799343522579037175014994154287033234499296875} a^{14} - \frac{1317128237786856363640974432160557163530070358}{1958996717612895185875074970771435166172496484375} a^{13} - \frac{842370142491996175896244583520662281516945804}{1958996717612895185875074970771435166172496484375} a^{12} - \frac{4170089147970403152068853020238021700974726587}{1958996717612895185875074970771435166172496484375} a^{11} - \frac{113426549081886097986392261981891271122794658357}{1958996717612895185875074970771435166172496484375} a^{10} - \frac{194333698039816303800064088225073840614064120771}{1958996717612895185875074970771435166172496484375} a^{9} - \frac{229063100548554754713288474005072158047594116323}{1958996717612895185875074970771435166172496484375} a^{8} + \frac{139036674856652345720136954486727826326215272604}{391799343522579037175014994154287033234499296875} a^{7} - \frac{7771701824760400799485393432359675888479450183}{78359868704515807435002998830857406646899859375} a^{6} + \frac{6218020478681279980916754351084846896993179196}{15671973740903161487000599766171481329379971875} a^{5} - \frac{522756534898750865959178996257590883314947993}{3134394748180632297400119953234296265875994375} a^{4} + \frac{107642665440848634908482593414841026966762899}{626878949636126459480023990646859253175198875} a^{3} - \frac{18870388388365666468225120538062512947884767}{125375789927225291896004798129371850635039775} a^{2} - \frac{1068028963695000040427292606128367605277220}{5015031597089011675840191925174874025401591} a + \frac{2133671292130538052081978149870616522689099}{5015031597089011675840191925174874025401591}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 1338628985.0976017 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times D_9:C_3$ (as 18T45):

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 $C_2\times D_9:C_3$
Character table for $C_2\times D_9:C_3$

Intermediate fields

\(\Q(\sqrt{69}) \), 3.1.108.1, 6.2.425747664.3, 9.1.11019960576.1

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

Sibling fields

Degree 18 sibling: 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ $18$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ $18$ R ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ $18$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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
2Data not computed
3Data not computed
$23$23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.12.6.1$x^{12} + 365010 x^{6} - 6436343 x^{2} + 33308075025$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$