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Properties

Label 18.0.59538292947...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{33}\cdot 5^{12}\cdot 7^{12}\cdot 29^{5}$
Root discriminant $97.16$
Ramified primes $2, 5, 7, 29$
Class number $98912$ (GRH)
Class group $[2, 2, 2, 12364]$ (GRH)
Galois group 18T176

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![437249189, -445836796, 444155462, -351569070, 205773548, -121071826, 57594463, -22885542, 9953429, -2750554, 978299, -220932, 52049, -11216, 2116, -270, 69, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 + 69*x^16 - 270*x^15 + 2116*x^14 - 11216*x^13 + 52049*x^12 - 220932*x^11 + 978299*x^10 - 2750554*x^9 + 9953429*x^8 - 22885542*x^7 + 57594463*x^6 - 121071826*x^5 + 205773548*x^4 - 351569070*x^3 + 444155462*x^2 - 445836796*x + 437249189)
 
gp: K = bnfinit(x^18 - 2*x^17 + 69*x^16 - 270*x^15 + 2116*x^14 - 11216*x^13 + 52049*x^12 - 220932*x^11 + 978299*x^10 - 2750554*x^9 + 9953429*x^8 - 22885542*x^7 + 57594463*x^6 - 121071826*x^5 + 205773548*x^4 - 351569070*x^3 + 444155462*x^2 - 445836796*x + 437249189, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{17} + 69 x^{16} - 270 x^{15} + 2116 x^{14} - 11216 x^{13} + 52049 x^{12} - 220932 x^{11} + 978299 x^{10} - 2750554 x^{9} + 9953429 x^{8} - 22885542 x^{7} + 57594463 x^{6} - 121071826 x^{5} + 205773548 x^{4} - 351569070 x^{3} + 444155462 x^{2} - 445836796 x + 437249189 \)

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:  \(-595382929470791769653248000000000000=-\,2^{33}\cdot 5^{12}\cdot 7^{12}\cdot 29^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $97.16$
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:  $2, 5, 7, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{4909148996332359334928418889193279734126766849017753742643631356067473571} a^{17} - \frac{1616125625911953466252940232163023994860734167490986455625347637618761675}{4909148996332359334928418889193279734126766849017753742643631356067473571} a^{16} - \frac{1190118428621509450916764925424890561677300604588685311804614346966882285}{4909148996332359334928418889193279734126766849017753742643631356067473571} a^{15} - \frac{153499853090155583723000893129422808611231214679113754194276666275208444}{4909148996332359334928418889193279734126766849017753742643631356067473571} a^{14} - \frac{1657896349068302617999310365224658925906645605743061764918572876989768845}{4909148996332359334928418889193279734126766849017753742643631356067473571} a^{13} - \frac{1536633581911447983226025792542883924553728231935741424384865949555312787}{4909148996332359334928418889193279734126766849017753742643631356067473571} a^{12} + \frac{256578770304203214358314508703240166200858385414496116743546953010983278}{4909148996332359334928418889193279734126766849017753742643631356067473571} a^{11} + \frac{928321071153008360974880179781247215368593673458694868074566500959857195}{4909148996332359334928418889193279734126766849017753742643631356067473571} a^{10} + \frac{1388381830301581503891671684224622709101568388009036251531945270725643407}{4909148996332359334928418889193279734126766849017753742643631356067473571} a^{9} + \frac{283058945183209413905473934404417001352899453104736674853463118522304231}{4909148996332359334928418889193279734126766849017753742643631356067473571} a^{8} - \frac{354090980933784605247274113348347770497620071933638779819980607600110}{4909148996332359334928418889193279734126766849017753742643631356067473571} a^{7} - \frac{2068074464280591875200105670480490782477877161067419195401812087860309265}{4909148996332359334928418889193279734126766849017753742643631356067473571} a^{6} + \frac{2031745279415620791450870663817744982422431697639420359358474416713457416}{4909148996332359334928418889193279734126766849017753742643631356067473571} a^{5} + \frac{1453469168084110765527849448729603657762116820442642580031966502600924581}{4909148996332359334928418889193279734126766849017753742643631356067473571} a^{4} + \frac{252404181871981180310035680092985827629016593324566995046568306372864519}{4909148996332359334928418889193279734126766849017753742643631356067473571} a^{3} - \frac{1782048036747508108598080936604202535648301024820530307005985419286532389}{4909148996332359334928418889193279734126766849017753742643631356067473571} a^{2} - \frac{2227632983349543871158126089040269247504817312826325763965300042409151878}{4909148996332359334928418889193279734126766849017753742643631356067473571} a - \frac{1487878437784670760965389441899354531257169181204960453692692542585966862}{4909148996332359334928418889193279734126766849017753742643631356067473571}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{12364}$, which has order $98912$ (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:  \( 252178.434124 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T176:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 576
The 40 conjugacy class representatives for t18n176
Character table for t18n176 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 3.3.9800.1, 9.9.941192000000.1

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 ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ R R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }$

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$2.6.9.8$x^{6} + 4 x^{2} - 24$$2$$3$$9$$A_4\times C_2$$[2, 2, 3]^{3}$
2.12.24.244$x^{12} - 8 x^{11} + 4 x^{10} - 8 x^{9} - 14 x^{8} + 8 x^{7} + 8 x^{5} + 16 x^{3} - 8 x^{2} + 16 x + 8$$4$$3$$24$$C_2^2 \times A_4$$[2, 2, 3]^{6}$
5Data not computed
7Data not computed
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$