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Properties

Label 18.4.32881052489...9939.1
Degree $18$
Signature $[4, 7]$
Discriminant $-\,3^{27}\cdot 73^{3}\cdot 577^{4}$
Root discriminant $43.63$
Ramified primes $3, 73, 577$
Class number $8$ (GRH)
Class group $[2, 4]$ (GRH)
Galois group 18T767

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![447697, -124860, -207960, 162203, -10557, -118608, 133855, -82590, 26325, 2806, -9780, 6810, -2804, 663, 0, -81, 36, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 36*x^16 - 81*x^15 + 663*x^13 - 2804*x^12 + 6810*x^11 - 9780*x^10 + 2806*x^9 + 26325*x^8 - 82590*x^7 + 133855*x^6 - 118608*x^5 - 10557*x^4 + 162203*x^3 - 207960*x^2 - 124860*x + 447697)
 
gp: K = bnfinit(x^18 - 9*x^17 + 36*x^16 - 81*x^15 + 663*x^13 - 2804*x^12 + 6810*x^11 - 9780*x^10 + 2806*x^9 + 26325*x^8 - 82590*x^7 + 133855*x^6 - 118608*x^5 - 10557*x^4 + 162203*x^3 - 207960*x^2 - 124860*x + 447697, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 36 x^{16} - 81 x^{15} + 663 x^{13} - 2804 x^{12} + 6810 x^{11} - 9780 x^{10} + 2806 x^{9} + 26325 x^{8} - 82590 x^{7} + 133855 x^{6} - 118608 x^{5} - 10557 x^{4} + 162203 x^{3} - 207960 x^{2} - 124860 x + 447697 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-328810524890493731499656799939=-\,3^{27}\cdot 73^{3}\cdot 577^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.63$
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:  $3, 73, 577$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{19} a^{16} + \frac{8}{19} a^{15} - \frac{1}{19} a^{14} + \frac{2}{19} a^{12} - \frac{6}{19} a^{11} - \frac{3}{19} a^{10} + \frac{7}{19} a^{9} - \frac{3}{19} a^{8} + \frac{5}{19} a^{7} + \frac{6}{19} a^{6} + \frac{7}{19} a^{4} - \frac{5}{19} a^{3} + \frac{3}{19} a^{2} + \frac{4}{19} a$, $\frac{1}{72421466480294482646558577814319139048876178327} a^{17} + \frac{20487657695362488911042473513325091977271562}{72421466480294482646558577814319139048876178327} a^{16} - \frac{34193003908714310056575393919600119747449093112}{72421466480294482646558577814319139048876178327} a^{15} - \frac{22951273304727474264305134180730705186503423014}{72421466480294482646558577814319139048876178327} a^{14} - \frac{15349652702740154590440501199156013452784917271}{72421466480294482646558577814319139048876178327} a^{13} - \frac{4788703653216406580118479681412796149342433215}{72421466480294482646558577814319139048876178327} a^{12} - \frac{32096887841855548245874279710248944899924053012}{72421466480294482646558577814319139048876178327} a^{11} - \frac{1814951732574252279307137253702753831292313227}{3811656130541814876134661990227323107835588333} a^{10} - \frac{10261181053351183825824410896214351239295765897}{72421466480294482646558577814319139048876178327} a^{9} + \frac{20341268614629373433414314091095896801255569261}{72421466480294482646558577814319139048876178327} a^{8} + \frac{8271591380474029622641976145462272719978069421}{72421466480294482646558577814319139048876178327} a^{7} + \frac{16867540317652464292229448583050928668451051940}{72421466480294482646558577814319139048876178327} a^{6} - \frac{3033873273636201337430388358206140161491501693}{72421466480294482646558577814319139048876178327} a^{5} + \frac{1538992839514353260197891778482431625856196027}{72421466480294482646558577814319139048876178327} a^{4} - \frac{2989110835071840876945525367128495426303438888}{72421466480294482646558577814319139048876178327} a^{3} - \frac{18634929182085705765579346909919115460604068407}{72421466480294482646558577814319139048876178327} a^{2} - \frac{2520119628606798495190037330819628973281247554}{72421466480294482646558577814319139048876178327} a - \frac{279481876399498681287691670057903559358593691}{3811656130541814876134661990227323107835588333}$

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

Class group and class number

$C_{2}\times C_{4}$, which has order $8$ (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:  $10$
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:  \( 3507229.71507 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T767:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 82944
The 110 conjugacy class representatives for t18n767 are not computed
Character table for t18n767 is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\), 9.9.22384826361.1

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 $18$ R ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ $18$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ $18$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ $18$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }$

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
3Data not computed
73Data not computed
577Data not computed