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Properties

Label 18.0.18466288295...6067.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{9}\cdot 9685993193^{2}$
Root discriminant $22.29$
Ramified primes $3, 9685993193$
Class number $6$
Class group $[6]$
Galois group 18T913

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, 11, -8, 56, -32, 176, -18, 290, -56, 268, -31, 153, -25, 54, -6, 11, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 + 11*x^16 - 6*x^15 + 54*x^14 - 25*x^13 + 153*x^12 - 31*x^11 + 268*x^10 - 56*x^9 + 290*x^8 - 18*x^7 + 176*x^6 - 32*x^5 + 56*x^4 - 8*x^3 + 11*x^2 - 2*x + 1)
 
gp: K = bnfinit(x^18 - 2*x^17 + 11*x^16 - 6*x^15 + 54*x^14 - 25*x^13 + 153*x^12 - 31*x^11 + 268*x^10 - 56*x^9 + 290*x^8 - 18*x^7 + 176*x^6 - 32*x^5 + 56*x^4 - 8*x^3 + 11*x^2 - 2*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{17} + 11 x^{16} - 6 x^{15} + 54 x^{14} - 25 x^{13} + 153 x^{12} - 31 x^{11} + 268 x^{10} - 56 x^{9} + 290 x^{8} - 18 x^{7} + 176 x^{6} - 32 x^{5} + 56 x^{4} - 8 x^{3} + 11 x^{2} - 2 x + 1 \)

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:  \(-1846628829566101684706067=-\,3^{9}\cdot 9685993193^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.29$
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:  $3, 9685993193$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{134084699660567} a^{17} + \frac{24723520300967}{134084699660567} a^{16} - \frac{35894007967652}{134084699660567} a^{15} - \frac{24142590660318}{134084699660567} a^{14} - \frac{1377599782747}{4623610333123} a^{13} - \frac{54928096848592}{134084699660567} a^{12} + \frac{32855126716157}{134084699660567} a^{11} + \frac{1030601181392}{7887335274151} a^{10} - \frac{9302637841972}{134084699660567} a^{9} - \frac{19714566605366}{134084699660567} a^{8} + \frac{36077805356797}{134084699660567} a^{7} + \frac{38792722482372}{134084699660567} a^{6} + \frac{3347961759300}{7887335274151} a^{5} + \frac{961746910291}{4623610333123} a^{4} - \frac{33819270942301}{134084699660567} a^{3} + \frac{6683010180612}{134084699660567} a^{2} + \frac{22053598063867}{134084699660567} a + \frac{33877305470133}{134084699660567}$

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

Class group and class number

$C_{6}$, which has order $6$

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:  \( \frac{47891583738958}{134084699660567} a^{17} - \frac{86820287060379}{134084699660567} a^{16} + \frac{501190544012592}{134084699660567} a^{15} - \frac{172963595098720}{134084699660567} a^{14} + \frac{84468781516528}{4623610333123} a^{13} - \frac{667875601807442}{134084699660567} a^{12} + \frac{6713292299222856}{134084699660567} a^{11} + \frac{5211716338185}{7887335274151} a^{10} + \frac{11499168703457146}{134084699660567} a^{9} - \frac{35183313368740}{134084699660567} a^{8} + \frac{11655336261862436}{134084699660567} a^{7} + \frac{2239951952777386}{134084699660567} a^{6} + \frac{386122982155276}{7887335274151} a^{5} + \frac{7083549191957}{4623610333123} a^{4} + \frac{1557602400289508}{134084699660567} a^{3} + \frac{540225205368244}{134084699660567} a^{2} + \frac{258855302843097}{134084699660567} a + \frac{108033799801208}{134084699660567} \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 15971.1124201 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T913:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 725760
The 60 conjugacy class representatives for t18n913 are not computed
Character table for t18n913 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 9.9.9685993193.1

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 $18$ R ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/11.14.0.1}{14} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.14.0.1}{14} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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
$3$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
9685993193Data not computed