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Properties

Label 18.0.21409961693...0528.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{36}\cdot 3^{9}\cdot 3547^{4}$
Root discriminant $42.61$
Ramified primes $2, 3, 3547$
Class number $79$ (GRH)
Class group $[79]$ (GRH)
Galois group 18T888

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9, -9, 105, -96, 904, -736, 3040, -648, 5853, -2429, 3453, -1088, 1208, -360, 232, -32, 17, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 + 17*x^16 - 32*x^15 + 232*x^14 - 360*x^13 + 1208*x^12 - 1088*x^11 + 3453*x^10 - 2429*x^9 + 5853*x^8 - 648*x^7 + 3040*x^6 - 736*x^5 + 904*x^4 - 96*x^3 + 105*x^2 - 9*x + 9)
 
gp: K = bnfinit(x^18 - x^17 + 17*x^16 - 32*x^15 + 232*x^14 - 360*x^13 + 1208*x^12 - 1088*x^11 + 3453*x^10 - 2429*x^9 + 5853*x^8 - 648*x^7 + 3040*x^6 - 736*x^5 + 904*x^4 - 96*x^3 + 105*x^2 - 9*x + 9, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} + 17 x^{16} - 32 x^{15} + 232 x^{14} - 360 x^{13} + 1208 x^{12} - 1088 x^{11} + 3453 x^{10} - 2429 x^{9} + 5853 x^{8} - 648 x^{7} + 3040 x^{6} - 736 x^{5} + 904 x^{4} - 96 x^{3} + 105 x^{2} - 9 x + 9 \)

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:  \(-214099616939181656131950870528=-\,2^{36}\cdot 3^{9}\cdot 3547^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.61$
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:  $2, 3, 3547$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a$, $\frac{1}{6} a^{14} + \frac{1}{6} a^{13} + \frac{1}{6} a^{10} - \frac{1}{6} a^{9} - \frac{1}{6} a^{8} + \frac{1}{6} a^{6} - \frac{1}{2} a^{5} - \frac{1}{6} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{2}$, $\frac{1}{6} a^{15} - \frac{1}{6} a^{13} + \frac{1}{6} a^{11} + \frac{1}{6} a^{10} + \frac{1}{6} a^{8} + \frac{1}{6} a^{7} - \frac{1}{6} a^{6} + \frac{1}{3} a^{5} - \frac{1}{6} a^{4} - \frac{1}{6} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{36} a^{16} + \frac{1}{18} a^{14} + \frac{1}{9} a^{12} - \frac{2}{9} a^{11} + \frac{1}{6} a^{10} + \frac{1}{9} a^{9} - \frac{5}{36} a^{8} + \frac{2}{9} a^{7} - \frac{5}{18} a^{6} + \frac{2}{9} a^{5} - \frac{1}{2} a^{4} - \frac{1}{9} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{4}$, $\frac{1}{8674890363643747412124} a^{17} - \frac{739982414118273299}{2891630121214582470708} a^{16} + \frac{342536023019444488237}{4337445181821873706062} a^{15} + \frac{59270404716465723557}{1445815060607291235354} a^{14} - \frac{476791697761675512860}{2168722590910936853031} a^{13} - \frac{743028429052822533217}{4337445181821873706062} a^{12} + \frac{119014489128488003503}{481938353535763745118} a^{11} + \frac{1019428205073307695905}{4337445181821873706062} a^{10} + \frac{750959529225306139495}{8674890363643747412124} a^{9} + \frac{1103866161179865851561}{8674890363643747412124} a^{8} + \frac{1596526698257086222285}{4337445181821873706062} a^{7} + \frac{1881113653040553187903}{4337445181821873706062} a^{6} - \frac{138056912571711806987}{1445815060607291235354} a^{5} + \frac{1250033167878804020677}{4337445181821873706062} a^{4} + \frac{3797395170017914388}{80323058922627290853} a^{3} - \frac{104028998918176543271}{722907530303645617677} a^{2} + \frac{28274257575796811023}{107097411896836387804} a - \frac{461171682844107275185}{963876707071527490236}$

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

Class group and class number

$C_{79}$, which has order $79$ (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:  \( -\frac{35150113431402124022}{2168722590910936853031} a^{17} + \frac{3120628688240893099}{722907530303645617677} a^{16} - \frac{567191510649760785856}{2168722590910936853031} a^{15} + \frac{227960646078973673584}{722907530303645617677} a^{14} - \frac{7255408529053081084400}{2168722590910936853031} a^{13} + \frac{6560358807639953833168}{2168722590910936853031} a^{12} - \frac{3578458023396277579952}{240969176767881872559} a^{11} + \frac{5953507782873495891184}{2168722590910936853031} a^{10} - \frac{88837227155074880826494}{2168722590910936853031} a^{9} - \frac{5843839427357390747014}{2168722590910936853031} a^{8} - \frac{130887366609100121493520}{2168722590910936853031} a^{7} - \frac{131765136696966058406368}{2168722590910936853031} a^{6} - \frac{23997456862515661384960}{722907530303645617677} a^{5} - \frac{43092990920676728786032}{2168722590910936853031} a^{4} - \frac{593520458337831826832}{240969176767881872559} a^{3} - \frac{7371717688224881552864}{722907530303645617677} a^{2} - \frac{22728573483095756546}{80323058922627290853} a - \frac{52774309891317739759}{240969176767881872559} \) (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:  \( 5284504.97814 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T888:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 362880
The 36 conjugacy class representatives for t18n888
Character table for t18n888 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 9.9.29682796068864.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

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 $18$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ $18$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.8.18.14$x^{8} + 2 x^{6} + 14 x^{4} + 4$$4$$2$$18$$(((C_4 \times C_2): C_2):C_2):C_2$$[2, 2, 3, 7/2, 7/2]^{2}$
2.8.18.14$x^{8} + 2 x^{6} + 14 x^{4} + 4$$4$$2$$18$$(((C_4 \times C_2): C_2):C_2):C_2$$[2, 2, 3, 7/2, 7/2]^{2}$
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3547Data not computed