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Properties

Label 18.0.45078101169...7843.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{9}\cdot 73^{12}$
Root discriminant $30.25$
Ramified primes $3, 73$
Class number $4$
Class group $[2, 2]$
Galois group $S_3 \times C_3$ (as 18T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![729, -243, 1296, 729, 2214, 243, 324, 288, 642, 306, -170, 15, 203, -69, -15, 12, 5, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 5*x^16 + 12*x^15 - 15*x^14 - 69*x^13 + 203*x^12 + 15*x^11 - 170*x^10 + 306*x^9 + 642*x^8 + 288*x^7 + 324*x^6 + 243*x^5 + 2214*x^4 + 729*x^3 + 1296*x^2 - 243*x + 729)
 
gp: K = bnfinit(x^18 - 3*x^17 + 5*x^16 + 12*x^15 - 15*x^14 - 69*x^13 + 203*x^12 + 15*x^11 - 170*x^10 + 306*x^9 + 642*x^8 + 288*x^7 + 324*x^6 + 243*x^5 + 2214*x^4 + 729*x^3 + 1296*x^2 - 243*x + 729, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 5 x^{16} + 12 x^{15} - 15 x^{14} - 69 x^{13} + 203 x^{12} + 15 x^{11} - 170 x^{10} + 306 x^{9} + 642 x^{8} + 288 x^{7} + 324 x^{6} + 243 x^{5} + 2214 x^{4} + 729 x^{3} + 1296 x^{2} - 243 x + 729 \)

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:  \(-450781011699067762218867843=-\,3^{9}\cdot 73^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.25$
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:  $3, 73$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{9} - \frac{1}{3} a^{5} - \frac{4}{9} a^{3} - \frac{1}{3} a$, $\frac{1}{9} a^{10} - \frac{1}{9} a^{4}$, $\frac{1}{9} a^{11} - \frac{1}{9} a^{5}$, $\frac{1}{9} a^{12} - \frac{1}{9} a^{6}$, $\frac{1}{27} a^{13} - \frac{1}{27} a^{11} + \frac{1}{9} a^{8} - \frac{4}{27} a^{7} + \frac{1}{9} a^{6} - \frac{11}{27} a^{5} + \frac{4}{9} a^{4} - \frac{1}{9} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{81} a^{14} - \frac{1}{81} a^{12} + \frac{1}{27} a^{9} + \frac{5}{81} a^{8} + \frac{4}{27} a^{7} - \frac{2}{81} a^{6} + \frac{7}{27} a^{5} + \frac{2}{27} a^{4} - \frac{1}{3} a^{3} + \frac{4}{9} a^{2} - \frac{1}{3} a$, $\frac{1}{43416} a^{15} - \frac{1}{21708} a^{14} - \frac{109}{43416} a^{13} + \frac{275}{10854} a^{12} - \frac{109}{4824} a^{11} + \frac{415}{14472} a^{10} - \frac{601}{10854} a^{9} - \frac{647}{21708} a^{8} + \frac{3025}{43416} a^{7} + \frac{13}{648} a^{6} - \frac{101}{536} a^{5} - \frac{77}{1809} a^{4} - \frac{1385}{4824} a^{3} + \frac{1393}{4824} a^{2} + \frac{605}{1608} a + \frac{195}{536}$, $\frac{1}{76976568} a^{16} + \frac{145}{25658856} a^{15} - \frac{406735}{76976568} a^{14} - \frac{266359}{25658856} a^{13} + \frac{906197}{25658856} a^{12} + \frac{85381}{6414714} a^{11} + \frac{1930973}{76976568} a^{10} - \frac{228169}{4276476} a^{9} - \frac{8105045}{76976568} a^{8} - \frac{414913}{6414714} a^{7} - \frac{103601}{4276476} a^{6} - \frac{441275}{2850984} a^{5} + \frac{82069}{2850984} a^{4} + \frac{28697}{237582} a^{3} - \frac{323159}{712746} a^{2} - \frac{189713}{475164} a + \frac{140965}{316776}$, $\frac{1}{690556791528} a^{17} - \frac{2279}{690556791528} a^{16} + \frac{707809}{86319598941} a^{15} + \frac{2238476999}{690556791528} a^{14} + \frac{432097427}{28773199647} a^{13} + \frac{338700031}{12788088732} a^{12} - \frac{4604525359}{172639197882} a^{11} - \frac{4951334491}{690556791528} a^{10} - \frac{6099089117}{690556791528} a^{9} + \frac{13007617412}{86319598941} a^{8} + \frac{15982974833}{230185597176} a^{7} + \frac{518364145}{12788088732} a^{6} + \frac{189362480}{1065674061} a^{5} - \frac{215952129}{473632916} a^{4} - \frac{3667073603}{25576177464} a^{3} + \frac{2360607337}{25576177464} a^{2} + \frac{882930043}{4262696244} a - \frac{11443421}{2841797496}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$

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:  \( -\frac{937579}{3505364424} a^{17} + \frac{5063}{97371234} a^{16} + \frac{3004237}{3505364424} a^{15} - \frac{437629}{64914156} a^{14} - \frac{6230657}{1168454808} a^{13} + \frac{34531697}{1168454808} a^{12} - \frac{817765}{438170553} a^{11} - \frac{43714975}{292113702} a^{10} + \frac{79631849}{3505364424} a^{9} + \frac{59625887}{1168454808} a^{8} - \frac{158813581}{389484936} a^{7} - \frac{8636273}{16228539} a^{6} - \frac{40873397}{129828312} a^{5} - \frac{13073}{215304} a^{4} - \frac{89573011}{129828312} a^{3} - \frac{73056277}{43276104} a^{2} - \frac{858260}{1803171} a + \frac{403491}{2404228} \) (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:  \( 11712449.488 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3$ (as 18T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 9 conjugacy class representatives for $S_3 \times C_3$
Character table for $S_3 \times C_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.15987.1 x3, 3.3.5329.1, 6.0.766752507.2, 6.0.143883.1 x2, 6.0.766752507.1, 9.3.4086024109803.1 x3

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

Sibling fields

Degree 6 sibling: 6.0.143883.1
Degree 9 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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$3.2.1.2$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}$
3.2.1.2$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}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
$73$73.3.2.1$x^{3} - 73$$3$$1$$2$$C_3$$[\ ]_{3}$
73.3.2.1$x^{3} - 73$$3$$1$$2$$C_3$$[\ ]_{3}$
73.3.2.1$x^{3} - 73$$3$$1$$2$$C_3$$[\ ]_{3}$
73.3.2.1$x^{3} - 73$$3$$1$$2$$C_3$$[\ ]_{3}$
73.3.2.1$x^{3} - 73$$3$$1$$2$$C_3$$[\ ]_{3}$
73.3.2.1$x^{3} - 73$$3$$1$$2$$C_3$$[\ ]_{3}$