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Properties

Label 18.18.2153372151...4000.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{12}\cdot 3^{28}\cdot 5^{3}\cdot 107^{9}$
Root discriminant $118.59$
Ramified primes $2, 3, 5, 107$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 18T367

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-748889, -9510045, -38221929, -50506104, 1448733, 46353909, 18694341, -14087169, -8873151, 1584665, 1674201, -144, -145764, -10131, 5862, 465, -117, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 - 117*x^16 + 465*x^15 + 5862*x^14 - 10131*x^13 - 145764*x^12 - 144*x^11 + 1674201*x^10 + 1584665*x^9 - 8873151*x^8 - 14087169*x^7 + 18694341*x^6 + 46353909*x^5 + 1448733*x^4 - 50506104*x^3 - 38221929*x^2 - 9510045*x - 748889)
 
gp: K = bnfinit(x^18 - 6*x^17 - 117*x^16 + 465*x^15 + 5862*x^14 - 10131*x^13 - 145764*x^12 - 144*x^11 + 1674201*x^10 + 1584665*x^9 - 8873151*x^8 - 14087169*x^7 + 18694341*x^6 + 46353909*x^5 + 1448733*x^4 - 50506104*x^3 - 38221929*x^2 - 9510045*x - 748889, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} - 117 x^{16} + 465 x^{15} + 5862 x^{14} - 10131 x^{13} - 145764 x^{12} - 144 x^{11} + 1674201 x^{10} + 1584665 x^{9} - 8873151 x^{8} - 14087169 x^{7} + 18694341 x^{6} + 46353909 x^{5} + 1448733 x^{4} - 50506104 x^{3} - 38221929 x^{2} - 9510045 x - 748889 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(21533721517796829699270378833524224000=2^{12}\cdot 3^{28}\cdot 5^{3}\cdot 107^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $118.59$
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:  $2, 3, 5, 107$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{9} + \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{10} + \frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{6} a^{14} - \frac{1}{6} a^{12} + \frac{1}{6} a^{11} - \frac{1}{2} a^{10} - \frac{1}{6} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{6} a^{2} - \frac{1}{2} a - \frac{1}{6}$, $\frac{1}{12} a^{15} - \frac{1}{12} a^{14} - \frac{1}{12} a^{13} - \frac{1}{6} a^{12} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{4} a^{7} - \frac{1}{6} a^{6} + \frac{1}{6} a^{5} + \frac{1}{6} a^{4} - \frac{5}{12} a^{3} - \frac{1}{3} a^{2} + \frac{1}{6} a + \frac{1}{4}$, $\frac{1}{24} a^{16} + \frac{1}{24} a^{13} - \frac{1}{6} a^{12} - \frac{1}{4} a^{11} - \frac{1}{2} a^{10} + \frac{1}{12} a^{9} + \frac{3}{8} a^{8} + \frac{1}{24} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} + \frac{1}{24} a^{4} - \frac{1}{24} a^{3} - \frac{1}{2} a^{2} + \frac{1}{8} a + \frac{5}{24}$, $\frac{1}{541292035562154748520648765733142096225051336142088} a^{17} - \frac{4985786182832034122076086537127586272769301135345}{270646017781077374260324382866571048112525668071044} a^{16} - \frac{3934694707233455371178794566525619598878546472177}{270646017781077374260324382866571048112525668071044} a^{15} + \frac{24612128025775628401139647195027385285685655907435}{541292035562154748520648765733142096225051336142088} a^{14} + \frac{328768302854013738886704674734018650852787187653}{45107669630179562376720730477761841352087611345174} a^{13} + \frac{972816910339360093776773901249748037839236141373}{90215339260359124753441460955523682704175222690348} a^{12} - \frac{643517270008488699105245452351674516901996434919}{67661504445269343565081095716642762028131417017761} a^{11} - \frac{35561672678304800301907920518471484899941346990131}{270646017781077374260324382866571048112525668071044} a^{10} - \frac{242007462105666562697068106551497693078728628563135}{541292035562154748520648765733142096225051336142088} a^{9} + \frac{264447822078973445839804457201883823619374472085235}{541292035562154748520648765733142096225051336142088} a^{8} + \frac{33195450526326894659887931516217494060673168655465}{67661504445269343565081095716642762028131417017761} a^{7} + \frac{32431370029676131928637918099311283836180336985907}{67661504445269343565081095716642762028131417017761} a^{6} + \frac{136083335606986124980490847197235439086135604120493}{541292035562154748520648765733142096225051336142088} a^{5} + \frac{29836056802861557082742415450161894095295277068491}{180430678520718249506882921911047365408350445380696} a^{4} - \frac{4149081739558036616334675322774125662310207039128}{22553834815089781188360365238880920676043805672587} a^{3} - \frac{259715902725124695026438434419192513012616091447213}{541292035562154748520648765733142096225051336142088} a^{2} + \frac{263098701552747435412257425082487447801989478502539}{541292035562154748520648765733142096225051336142088} a - \frac{17500750707681725389540116707680228406366278892032}{67661504445269343565081095716642762028131417017761}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $17$
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:  \( 7755126924130 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T367:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2304
The 48 conjugacy class representatives for t18n367
Character table for t18n367 is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\), 3.3.321.1, 9.9.1953114230889.1

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 R ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.6.0.1}{6} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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$2.6.6.6$x^{6} - 13 x^{4} + 7 x^{2} - 3$$2$$3$$6$$A_4\times C_2$$[2, 2, 2]^{3}$
2.6.6.1$x^{6} + x^{2} - 1$$2$$3$$6$$A_4$$[2, 2]^{3}$
2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
$3$3.6.9.3$x^{6} + 3 x^{4} + 24$$6$$1$$9$$C_6$$[2]_{2}$
3.12.19.43$x^{12} + 3 x^{10} - 3 x^{9} - 3 x^{8} - 3 x^{6} + 3 x^{3} + 3$$12$$1$$19$$D_4 \times C_3$$[2]_{4}^{2}$
$5$5.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
5.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
107Data not computed