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Properties

Label 18.0.56986226150...6707.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{27}\cdot 73^{3}\cdot 577^{3}$
Root discriminant $30.65$
Ramified primes $3, 73, 577$
Class number $24$
Class group $[2, 12]$
Galois group 18T285

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![361, -513, 3006, -2606, 6450, -2424, 2869, -1116, 1740, -871, 966, -459, 256, -75, 60, -18, 6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 6*x^16 - 18*x^15 + 60*x^14 - 75*x^13 + 256*x^12 - 459*x^11 + 966*x^10 - 871*x^9 + 1740*x^8 - 1116*x^7 + 2869*x^6 - 2424*x^5 + 6450*x^4 - 2606*x^3 + 3006*x^2 - 513*x + 361)
 
gp: K = bnfinit(x^18 + 6*x^16 - 18*x^15 + 60*x^14 - 75*x^13 + 256*x^12 - 459*x^11 + 966*x^10 - 871*x^9 + 1740*x^8 - 1116*x^7 + 2869*x^6 - 2424*x^5 + 6450*x^4 - 2606*x^3 + 3006*x^2 - 513*x + 361, 1)
 

Normalized defining polynomial

\( x^{18} + 6 x^{16} - 18 x^{15} + 60 x^{14} - 75 x^{13} + 256 x^{12} - 459 x^{11} + 966 x^{10} - 871 x^{9} + 1740 x^{8} - 1116 x^{7} + 2869 x^{6} - 2424 x^{5} + 6450 x^{4} - 2606 x^{3} + 3006 x^{2} - 513 x + 361 \)

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:  \(-569862261508654647313096707=-\,3^{27}\cdot 73^{3}\cdot 577^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.65$
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 a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{2993663730364181384005520607601} a^{17} - \frac{47042378421876069378981061944}{157561248966535862316080031979} a^{16} + \frac{565504415408040083261445567062}{2993663730364181384005520607601} a^{15} - \frac{926884006177791023007230426673}{2993663730364181384005520607601} a^{14} - \frac{352076789582274912894166781641}{2993663730364181384005520607601} a^{13} + \frac{335403059131488635365336644565}{2993663730364181384005520607601} a^{12} + \frac{1103180718421321636425626301951}{2993663730364181384005520607601} a^{11} - \frac{340238561233270490158820650561}{2993663730364181384005520607601} a^{10} + \frac{391372080934192337727560379191}{2993663730364181384005520607601} a^{9} + \frac{791269858585048479405768966005}{2993663730364181384005520607601} a^{8} - \frac{4265484652410841670794368043}{176097866492010669647383565153} a^{7} - \frac{455102135054269215733545812556}{2993663730364181384005520607601} a^{6} + \frac{4375040032785130830151037878}{157561248966535862316080031979} a^{5} - \frac{1488134829599160309509297384367}{2993663730364181384005520607601} a^{4} + \frac{1338248059913619262888092915483}{2993663730364181384005520607601} a^{3} - \frac{479052853161200656390273438266}{2993663730364181384005520607601} a^{2} - \frac{68362412065054888691049520700}{2993663730364181384005520607601} a - \frac{58281876123003494548604157719}{157561248966535862316080031979}$

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

Class group and class number

$C_{2}\times C_{12}$, which has order $24$

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:  \( -1 \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 26510.9946997 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T285:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1296
The 34 conjugacy class representatives for t18n285
Character table for t18n285 is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\), 6.0.829067643.1, 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 $18$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ $18$ $18$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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