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Properties

Label 18.0.17011815946...7968.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{16}\cdot 3^{37}\cdot 7^{8}$
Root discriminant $42.07$
Ramified primes $2, 3, 7$
Class number $10$ (GRH)
Class group $[10]$ (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![1936, 6336, 20736, 21120, 34560, 4752, 42048, -792, 20736, -88, 7200, 0, 1464, 0, 216, 0, 18, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 18*x^16 + 216*x^14 + 1464*x^12 + 7200*x^10 - 88*x^9 + 20736*x^8 - 792*x^7 + 42048*x^6 + 4752*x^5 + 34560*x^4 + 21120*x^3 + 20736*x^2 + 6336*x + 1936)
 
gp: K = bnfinit(x^18 + 18*x^16 + 216*x^14 + 1464*x^12 + 7200*x^10 - 88*x^9 + 20736*x^8 - 792*x^7 + 42048*x^6 + 4752*x^5 + 34560*x^4 + 21120*x^3 + 20736*x^2 + 6336*x + 1936, 1)
 

Normalized defining polynomial

\( x^{18} + 18 x^{16} + 216 x^{14} + 1464 x^{12} + 7200 x^{10} - 88 x^{9} + 20736 x^{8} - 792 x^{7} + 42048 x^{6} + 4752 x^{5} + 34560 x^{4} + 21120 x^{3} + 20736 x^{2} + 6336 x + 1936 \)

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:  \(-170118159464158166333506387968=-\,2^{16}\cdot 3^{37}\cdot 7^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.07$
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:  $2, 3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{36} a^{9} + \frac{1}{3} a^{3} - \frac{2}{9}$, $\frac{1}{36} a^{10} + \frac{1}{3} a^{4} - \frac{2}{9} a$, $\frac{1}{72} a^{11} + \frac{1}{6} a^{5} + \frac{7}{18} a^{2}$, $\frac{1}{72} a^{12} + \frac{1}{6} a^{6} + \frac{7}{18} a^{3}$, $\frac{1}{72} a^{13} + \frac{1}{6} a^{7} + \frac{7}{18} a^{4}$, $\frac{1}{216} a^{14} - \frac{1}{216} a^{13} + \frac{1}{216} a^{12} + \frac{1}{216} a^{11} + \frac{1}{108} a^{10} - \frac{1}{108} a^{9} + \frac{2}{9} a^{8} - \frac{2}{9} a^{7} + \frac{1}{18} a^{6} + \frac{1}{54} a^{5} - \frac{19}{54} a^{4} + \frac{1}{54} a^{3} - \frac{11}{54} a^{2} - \frac{11}{27} a - \frac{7}{27}$, $\frac{1}{216} a^{15} - \frac{1}{216} a^{12} - \frac{1}{108} a^{9} - \frac{1}{6} a^{7} - \frac{5}{54} a^{6} - \frac{1}{3} a^{4} - \frac{13}{54} a^{3} + \frac{1}{3} a - \frac{13}{27}$, $\frac{1}{432} a^{16} + \frac{1}{216} a^{13} + \frac{1}{108} a^{10} + \frac{1}{6} a^{8} - \frac{23}{108} a^{7} - \frac{1}{6} a^{5} + \frac{13}{54} a^{4} - \frac{1}{3} a^{2} + \frac{4}{27} a$, $\frac{1}{29760938172024048} a^{17} + \frac{744993521113}{2705539833820368} a^{16} + \frac{27604172412199}{14880469086012024} a^{15} - \frac{250341215479}{338192479227546} a^{14} - \frac{79481505603667}{14880469086012024} a^{13} - \frac{577321672447}{169096239613773} a^{12} + \frac{67062194127473}{14880469086012024} a^{11} + \frac{838837004033}{338192479227546} a^{10} - \frac{9428044721338}{1860058635751503} a^{9} + \frac{31030372172153}{676384958455092} a^{8} - \frac{1536633099616315}{7440234543006012} a^{7} + \frac{37513279717283}{338192479227546} a^{6} - \frac{394532334176147}{3720117271503006} a^{5} - \frac{84246994963259}{338192479227546} a^{4} - \frac{709761233521490}{1860058635751503} a^{3} - \frac{25646833478969}{338192479227546} a^{2} + \frac{452590013182415}{1860058635751503} a - \frac{23373242082566}{169096239613773}$

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

Class group and class number

$C_{10}$, which has order $10$ (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:  \( -\frac{188003832685}{1240039090501002} a^{17} - \frac{494168063}{25051294757596} a^{16} - \frac{188718132832}{68891060583389} a^{15} - \frac{69317193511}{225461652818364} a^{14} - \frac{2268140293362}{68891060583389} a^{13} - \frac{129687355069}{37576942136394} a^{12} - \frac{1111153219346569}{4960156362004008} a^{11} - \frac{121007652498}{6262823689399} a^{10} - \frac{456611876599369}{413346363500334} a^{9} - \frac{3795507719980}{56365413204591} a^{8} - \frac{220714992652996}{68891060583389} a^{7} + \frac{48395308844}{6262823689399} a^{6} - \frac{4042412486584796}{620019545250501} a^{5} - \frac{4822574801328}{6262823689399} a^{4} - \frac{1172233595264278}{206673181750167} a^{3} - \frac{250961820587063}{112730826409182} a^{2} - \frac{237600249964644}{68891060583389} a - \frac{1082590866313}{18788471068197} \) (order $6$)
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:  \( 12312115.5749 \) (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

\(\Q(\sqrt{-3}) \), 3.3.756.1, 6.0.1714608.1, 9.9.238130328086784.1

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 $18$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ $18$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ $18$

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
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$