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Properties

Label 18.0.25066383071...8467.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,7^{12}\cdot 1399^{2}\cdot 4523^{3}$
Root discriminant $33.28$
Ramified primes $7, 1399, 4523$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group 18T472

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![912673, -611585, 50634, 235477, -9158, -64228, 72116, -16179, 12191, -8311, 6431, -2823, 1149, -362, 184, -60, 27, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 + 27*x^16 - 60*x^15 + 184*x^14 - 362*x^13 + 1149*x^12 - 2823*x^11 + 6431*x^10 - 8311*x^9 + 12191*x^8 - 16179*x^7 + 72116*x^6 - 64228*x^5 - 9158*x^4 + 235477*x^3 + 50634*x^2 - 611585*x + 912673)
 
gp: K = bnfinit(x^18 - 4*x^17 + 27*x^16 - 60*x^15 + 184*x^14 - 362*x^13 + 1149*x^12 - 2823*x^11 + 6431*x^10 - 8311*x^9 + 12191*x^8 - 16179*x^7 + 72116*x^6 - 64228*x^5 - 9158*x^4 + 235477*x^3 + 50634*x^2 - 611585*x + 912673, 1)
 

Normalized defining polynomial

\( x^{18} - 4 x^{17} + 27 x^{16} - 60 x^{15} + 184 x^{14} - 362 x^{13} + 1149 x^{12} - 2823 x^{11} + 6431 x^{10} - 8311 x^{9} + 12191 x^{8} - 16179 x^{7} + 72116 x^{6} - 64228 x^{5} - 9158 x^{4} + 235477 x^{3} + 50634 x^{2} - 611585 x + 912673 \)

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

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:  \(-2506638307140843254915898467=-\,7^{12}\cdot 1399^{2}\cdot 4523^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.28$
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:  $7, 1399, 4523$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{97} a^{16} - \frac{4}{97} a^{15} + \frac{27}{97} a^{14} + \frac{37}{97} a^{13} - \frac{10}{97} a^{12} + \frac{26}{97} a^{11} - \frac{15}{97} a^{10} - \frac{10}{97} a^{9} + \frac{29}{97} a^{8} + \frac{31}{97} a^{7} - \frac{31}{97} a^{6} + \frac{20}{97} a^{5} + \frac{45}{97} a^{4} - \frac{14}{97} a^{3} - \frac{40}{97} a^{2} - \frac{39}{97} a$, $\frac{1}{30628175132447925323718875972005191049696966789829} a^{17} + \frac{1833282544583134435921021484354728815201519873}{1056143970084411218059271585241558312058516096201} a^{16} + \frac{1037170711669429100829603725762510790172862492997}{2356013471726763486439913536308091619207458983833} a^{15} - \frac{14773653966510485079669513960399430017201445058646}{30628175132447925323718875972005191049696966789829} a^{14} + \frac{14373196985888694969172216260321042608468476379978}{30628175132447925323718875972005191049696966789829} a^{13} - \frac{13445305649876079023800465214331585693170858219820}{30628175132447925323718875972005191049696966789829} a^{12} + \frac{5872875975572791453666553212168121903667096259794}{30628175132447925323718875972005191049696966789829} a^{11} + \frac{10919372635158816423597347098314515782309490835364}{30628175132447925323718875972005191049696966789829} a^{10} - \frac{696633473834950310158115327521743625634213981427}{2356013471726763486439913536308091619207458983833} a^{9} + \frac{3401503976467758131070296132348410953654882880577}{30628175132447925323718875972005191049696966789829} a^{8} - \frac{3294773473911318001842078329587326193363918899771}{30628175132447925323718875972005191049696966789829} a^{7} - \frac{957743467841496765426424994003137089745326284629}{2356013471726763486439913536308091619207458983833} a^{6} - \frac{10909875234399349636864552653785538495572900025605}{30628175132447925323718875972005191049696966789829} a^{5} - \frac{11987282003204743103908418722046687902431700308}{30628175132447925323718875972005191049696966789829} a^{4} - \frac{5884505495936980748274301492629990922135478311704}{30628175132447925323718875972005191049696966789829} a^{3} - \frac{13340492217226978810307906759432434307663267450542}{30628175132447925323718875972005191049696966789829} a^{2} - \frac{103184056495171772230731089679043321941881032645}{315754382808741498182668824453661763398937801957} a + \frac{86933188100297156840300407746152116803925385}{3255199822770530909099678602615069725762245381}$

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:  $8$
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:  \( 471320.757853 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T472:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5184
The 88 conjugacy class representatives for t18n472 are not computed
Character table for t18n472 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 6.0.10859723.1, 9.3.164590951.1

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 $18$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
7Data not computed
1399Data not computed
4523Data not computed