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Properties

Label 18.2.21795504923...8064.1
Degree $18$
Signature $[2, 8]$
Discriminant $2^{12}\cdot 3^{45}\cdot 23^{9}$
Root discriminant $118.67$
Ramified primes $2, 3, 23$
Class number $12$ (GRH)
Class group $[12]$ (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![-871019771, 393171066, 446766822, -233119176, -64912734, 40765104, -162705, 892350, -135864, -480272, 60390, 17460, 2499, 1116, -558, -60, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 60*x^15 - 558*x^14 + 1116*x^13 + 2499*x^12 + 17460*x^11 + 60390*x^10 - 480272*x^9 - 135864*x^8 + 892350*x^7 - 162705*x^6 + 40765104*x^5 - 64912734*x^4 - 233119176*x^3 + 446766822*x^2 + 393171066*x - 871019771)
 
gp: K = bnfinit(x^18 - 60*x^15 - 558*x^14 + 1116*x^13 + 2499*x^12 + 17460*x^11 + 60390*x^10 - 480272*x^9 - 135864*x^8 + 892350*x^7 - 162705*x^6 + 40765104*x^5 - 64912734*x^4 - 233119176*x^3 + 446766822*x^2 + 393171066*x - 871019771, 1)
 

Normalized defining polynomial

\( x^{18} - 60 x^{15} - 558 x^{14} + 1116 x^{13} + 2499 x^{12} + 17460 x^{11} + 60390 x^{10} - 480272 x^{9} - 135864 x^{8} + 892350 x^{7} - 162705 x^{6} + 40765104 x^{5} - 64912734 x^{4} - 233119176 x^{3} + 446766822 x^{2} + 393171066 x - 871019771 \)

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:  \(21795504923434979455845775529962328064=2^{12}\cdot 3^{45}\cdot 23^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $118.67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{6} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{6}$, $\frac{1}{6} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{6} a$, $\frac{1}{6} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{6} a^{2}$, $\frac{1}{6} a^{12} + \frac{1}{3} a^{3} - \frac{1}{2}$, $\frac{1}{6} a^{13} + \frac{1}{3} a^{4} - \frac{1}{2} a$, $\frac{1}{18} a^{14} + \frac{1}{18} a^{13} + \frac{1}{18} a^{12} - \frac{1}{18} a^{11} - \frac{1}{18} a^{10} - \frac{1}{18} a^{9} - \frac{1}{6} a^{8} - \frac{1}{6} a^{7} - \frac{1}{6} a^{6} - \frac{1}{18} a^{5} - \frac{1}{18} a^{4} - \frac{1}{18} a^{3} + \frac{2}{9} a^{2} + \frac{2}{9} a + \frac{2}{9}$, $\frac{1}{18} a^{15} + \frac{1}{18} a^{12} + \frac{1}{18} a^{9} - \frac{7}{18} a^{6} + \frac{1}{9} a^{3} + \frac{1}{9}$, $\frac{1}{2628} a^{16} + \frac{37}{2628} a^{15} + \frac{13}{1314} a^{14} - \frac{19}{292} a^{13} + \frac{11}{292} a^{12} - \frac{11}{657} a^{11} + \frac{11}{657} a^{10} - \frac{19}{657} a^{9} + \frac{137}{438} a^{8} - \frac{155}{1314} a^{7} - \frac{307}{657} a^{6} + \frac{232}{657} a^{5} + \frac{325}{876} a^{4} - \frac{107}{876} a^{3} + \frac{8}{657} a^{2} + \frac{757}{2628} a - \frac{1283}{2628}$, $\frac{1}{4576297300443344871891800522230421354977208664426317431236444} a^{17} - \frac{150575808218686922559389726422045786437794386702755889697}{1525432433481114957297266840743473784992402888142105810412148} a^{16} - \frac{18943214816052733294625570117533834583952261136709243818793}{1144074325110836217972950130557605338744302166106579357809111} a^{15} + \frac{74568931108447612996131640139881722648521602014866833580005}{4576297300443344871891800522230421354977208664426317431236444} a^{14} + \frac{2161935437234287740872132381664722848170488748268344483509}{508477477827038319099088946914491261664134296047368603470716} a^{13} - \frac{66419076307221456076069906973783222153550847129191737552099}{1144074325110836217972950130557605338744302166106579357809111} a^{12} - \frac{99292669416182583320359967725025908589055031478385416573029}{2288148650221672435945900261115210677488604332213158715618222} a^{11} + \frac{11124861745438799155611423963350876252402690697769605492713}{381358108370278739324316710185868446248100722035526452603037} a^{10} + \frac{12455690129667077689745314213602417752058838437834046022249}{2288148650221672435945900261115210677488604332213158715618222} a^{9} - \frac{57058116509003019122189203987048046565699356758001687883718}{1144074325110836217972950130557605338744302166106579357809111} a^{8} + \frac{171469695574987821308546142528080434737511610486651149820747}{381358108370278739324316710185868446248100722035526452603037} a^{7} + \frac{129598419290789836217707728475516393442273934970186408418096}{1144074325110836217972950130557605338744302166106579357809111} a^{6} - \frac{1713265550161899056069593135887972927416606376913003511547787}{4576297300443344871891800522230421354977208664426317431236444} a^{5} + \frac{27842067136848013086779125283980564969323783798065323874979}{508477477827038319099088946914491261664134296047368603470716} a^{4} - \frac{202087674098169122326433851253292379386489214623144803169323}{2288148650221672435945900261115210677488604332213158715618222} a^{3} - \frac{2167186844753692848610517538671031046329013110462363306299461}{4576297300443344871891800522230421354977208664426317431236444} a^{2} - \frac{209674613169458450697313911430814125758469697419424724399251}{1525432433481114957297266840743473784992402888142105810412148} a + \frac{17017275539442178842533354134214139691447929493663144507301}{1144074325110836217972950130557605338744302166106579357809111}$

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

Class group and class number

$C_{12}$, which has order $12$ (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:  \( 117503877267.76468 \) (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

\(\Q(\sqrt{69}) \), 3.1.243.1, 6.2.2155347549.1, 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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
$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}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
3Data not computed
$23$23.2.1.1$x^{2} - 23$$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}$