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Properties

Label 18.0.59170638284...0087.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{9}\cdot 7^{12}\cdot 109^{9}$
Root discriminant $66.17$
Ramified primes $3, 7, 109$
Class number $1296$ (GRH)
Class group $[2, 18, 36]$ (GRH)
Galois group $C_3^2 : C_2$ (as 18T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![36072, -55368, 27498, 122397, -234155, 192540, 59180, -303784, 328020, -178658, 52764, -17418, 9624, -3080, 468, -52, 32, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 32*x^16 - 52*x^15 + 468*x^14 - 3080*x^13 + 9624*x^12 - 17418*x^11 + 52764*x^10 - 178658*x^9 + 328020*x^8 - 303784*x^7 + 59180*x^6 + 192540*x^5 - 234155*x^4 + 122397*x^3 + 27498*x^2 - 55368*x + 36072)
 
gp: K = bnfinit(x^18 - 9*x^17 + 32*x^16 - 52*x^15 + 468*x^14 - 3080*x^13 + 9624*x^12 - 17418*x^11 + 52764*x^10 - 178658*x^9 + 328020*x^8 - 303784*x^7 + 59180*x^6 + 192540*x^5 - 234155*x^4 + 122397*x^3 + 27498*x^2 - 55368*x + 36072, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 32 x^{16} - 52 x^{15} + 468 x^{14} - 3080 x^{13} + 9624 x^{12} - 17418 x^{11} + 52764 x^{10} - 178658 x^{9} + 328020 x^{8} - 303784 x^{7} + 59180 x^{6} + 192540 x^{5} - 234155 x^{4} + 122397 x^{3} + 27498 x^{2} - 55368 x + 36072 \)

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:  \(-591706382841388634460616841610087=-\,3^{9}\cdot 7^{12}\cdot 109^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $66.17$
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:  $3, 7, 109$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{6} a^{8} + \frac{1}{3} a^{2} - \frac{1}{2} a$, $\frac{1}{18} a^{9} + \frac{1}{3} a^{5} - \frac{2}{9} a^{3} - \frac{1}{2} a^{2} + \frac{1}{3} a$, $\frac{1}{18} a^{10} + \frac{4}{9} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{18} a^{11} + \frac{4}{9} a^{5} - \frac{1}{2} a^{4}$, $\frac{1}{54} a^{12} - \frac{1}{54} a^{10} + \frac{1}{18} a^{8} - \frac{1}{9} a^{7} - \frac{2}{27} a^{6} - \frac{5}{18} a^{5} + \frac{8}{27} a^{4} + \frac{1}{18} a^{3} - \frac{4}{9} a^{2} - \frac{1}{2} a$, $\frac{1}{54} a^{13} - \frac{1}{54} a^{11} + \frac{1}{18} a^{8} - \frac{2}{27} a^{7} + \frac{1}{18} a^{6} - \frac{1}{27} a^{5} + \frac{7}{18} a^{4} - \frac{2}{9} a^{3} - \frac{1}{3} a^{2} + \frac{1}{6} a$, $\frac{1}{2430} a^{14} - \frac{7}{2430} a^{13} - \frac{2}{405} a^{12} + \frac{14}{1215} a^{11} + \frac{1}{486} a^{10} - \frac{11}{405} a^{9} - \frac{47}{1215} a^{8} - \frac{389}{2430} a^{7} - \frac{11}{810} a^{6} + \frac{463}{1215} a^{5} + \frac{40}{243} a^{4} - \frac{59}{810} a^{3} + \frac{23}{405} a^{2} - \frac{8}{27} a - \frac{7}{45}$, $\frac{1}{2430} a^{15} - \frac{8}{1215} a^{13} - \frac{11}{2430} a^{12} + \frac{7}{810} a^{11} + \frac{59}{2430} a^{10} - \frac{8}{1215} a^{9} + \frac{11}{810} a^{8} + \frac{17}{1215} a^{7} - \frac{16}{243} a^{6} - \frac{211}{810} a^{5} + \frac{913}{2430} a^{4} + \frac{263}{810} a^{3} - \frac{139}{405} a^{2} - \frac{31}{135} a - \frac{4}{45}$, $\frac{1}{32114729682741780} a^{16} - \frac{2}{8028682420685445} a^{15} + \frac{100004911181}{3211472968274178} a^{14} - \frac{233344792751}{1070490989424726} a^{13} - \frac{10393307442992}{8028682420685445} a^{12} + \frac{170221923902167}{16057364841370890} a^{11} - \frac{122441510780147}{8028682420685445} a^{10} - \frac{41937329137640}{1605736484137089} a^{9} + \frac{53688774306383}{16057364841370890} a^{8} - \frac{345174036727271}{5352454947123630} a^{7} - \frac{2523824068097273}{16057364841370890} a^{6} - \frac{16232873463062}{422562232667655} a^{5} - \frac{72223303312775}{3211472968274178} a^{4} + \frac{93874770111044}{297358608173535} a^{3} - \frac{3374101505168711}{10704909894247260} a^{2} + \frac{554817659460611}{1784151649041210} a + \frac{153065644144}{356118093621}$, $\frac{1}{44414671151231881740} a^{17} + \frac{683}{44414671151231881740} a^{16} - \frac{1158430568847319}{7402445191871980290} a^{15} - \frac{503004447073801}{11103667787807970435} a^{14} + \frac{66839074745976589}{11103667787807970435} a^{13} + \frac{1699876412478775}{493496346124798686} a^{12} + \frac{3224457892685909}{274164636735999270} a^{11} + \frac{8024392678137076}{411246955103998905} a^{10} - \frac{89164671269874527}{3701222595935990145} a^{9} + \frac{1135517138796357149}{22207335575615940870} a^{8} + \frac{366084647769788740}{2220733557561594087} a^{7} - \frac{4361800139234927}{411246955103998905} a^{6} + \frac{4563430367572079671}{22207335575615940870} a^{5} - \frac{7346363817966094453}{22207335575615940870} a^{4} + \frac{1009970115033458821}{14804890383743960580} a^{3} + \frac{2063203475453808199}{14804890383743960580} a^{2} - \frac{183137464995679904}{1233740865311996715} a + \frac{255501379324868}{2462556617389215}$

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

Class group and class number

$C_{2}\times C_{18}\times C_{36}$, which has order $1296$ (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:  \( -1 \) (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:  \( 40248551.3965 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3:S_3$ (as 18T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 6 conjugacy class representatives for $C_3^2 : C_2$
Character table for $C_3^2 : C_2$

Intermediate fields

\(\Q(\sqrt{-327}) \), 3.1.16023.1 x3, 3.1.16023.3 x3, 3.1.327.1 x3, 3.1.16023.2 x3, 6.0.83952844983.1, 6.0.83952844983.3, 6.0.34965783.1, 6.0.83952844983.2, 9.1.1345176435162609.4 x9

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

Sibling fields

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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/5.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$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$$[\ ]_{2}$
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
$109$109.2.1.1$x^{2} - 109$$2$$1$$1$$C_2$$[\ ]_{2}$
109.2.1.1$x^{2} - 109$$2$$1$$1$$C_2$$[\ ]_{2}$
109.2.1.1$x^{2} - 109$$2$$1$$1$$C_2$$[\ ]_{2}$
109.2.1.1$x^{2} - 109$$2$$1$$1$$C_2$$[\ ]_{2}$
109.2.1.1$x^{2} - 109$$2$$1$$1$$C_2$$[\ ]_{2}$
109.2.1.1$x^{2} - 109$$2$$1$$1$$C_2$$[\ ]_{2}$
109.2.1.1$x^{2} - 109$$2$$1$$1$$C_2$$[\ ]_{2}$
109.2.1.1$x^{2} - 109$$2$$1$$1$$C_2$$[\ ]_{2}$
109.2.1.1$x^{2} - 109$$2$$1$$1$$C_2$$[\ ]_{2}$