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Properties

Label 18.0.50078803235...8288.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{21}\cdot 43^{8}$
Root discriminant $30.43$
Ramified primes $2, 3, 43$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3^2:S_3$ (as 18T24)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3481, 885, 6420, -7947, 8091, -9051, 6691, -5790, 3147, -1691, 1230, -348, 268, -81, 42, -10, 9, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 9*x^16 - 10*x^15 + 42*x^14 - 81*x^13 + 268*x^12 - 348*x^11 + 1230*x^10 - 1691*x^9 + 3147*x^8 - 5790*x^7 + 6691*x^6 - 9051*x^5 + 8091*x^4 - 7947*x^3 + 6420*x^2 + 885*x + 3481)
 
gp: K = bnfinit(x^18 + 9*x^16 - 10*x^15 + 42*x^14 - 81*x^13 + 268*x^12 - 348*x^11 + 1230*x^10 - 1691*x^9 + 3147*x^8 - 5790*x^7 + 6691*x^6 - 9051*x^5 + 8091*x^4 - 7947*x^3 + 6420*x^2 + 885*x + 3481, 1)
 

Normalized defining polynomial

\( x^{18} + 9 x^{16} - 10 x^{15} + 42 x^{14} - 81 x^{13} + 268 x^{12} - 348 x^{11} + 1230 x^{10} - 1691 x^{9} + 3147 x^{8} - 5790 x^{7} + 6691 x^{6} - 9051 x^{5} + 8091 x^{4} - 7947 x^{3} + 6420 x^{2} + 885 x + 3481 \)

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:  \(-500788032352702912536588288=-\,2^{12}\cdot 3^{21}\cdot 43^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.43$
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:  $2, 3, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{6} - \frac{1}{3}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{7} - \frac{1}{3} a$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{8} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{6} - \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{7} - \frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{8} - \frac{1}{3} a^{5} + \frac{1}{3} a^{2}$, $\frac{1}{387} a^{15} + \frac{20}{387} a^{14} - \frac{11}{387} a^{13} + \frac{29}{387} a^{12} + \frac{5}{129} a^{11} - \frac{11}{387} a^{10} + \frac{17}{387} a^{9} - \frac{50}{387} a^{8} + \frac{46}{129} a^{7} - \frac{35}{129} a^{6} - \frac{179}{387} a^{5} - \frac{85}{387} a^{4} - \frac{70}{387} a^{3} + \frac{68}{387} a^{2} - \frac{59}{129} a - \frac{16}{387}$, $\frac{1}{120391443} a^{16} + \frac{36400}{40130481} a^{15} - \frac{2488709}{40130481} a^{14} + \frac{6127900}{40130481} a^{13} + \frac{626870}{120391443} a^{12} + \frac{3783475}{120391443} a^{11} - \frac{1215515}{40130481} a^{10} - \frac{5464024}{40130481} a^{9} + \frac{16542604}{120391443} a^{8} - \frac{6063281}{13376827} a^{7} - \frac{23446373}{120391443} a^{6} - \frac{5195195}{40130481} a^{5} - \frac{15320795}{120391443} a^{4} + \frac{1533496}{120391443} a^{3} + \frac{57857387}{120391443} a^{2} + \frac{4290122}{120391443} a + \frac{16411160}{120391443}$, $\frac{1}{6605656525298614423257} a^{17} - \frac{46954080043}{12440031121089669347} a^{16} + \frac{5458684610305023617}{6605656525298614423257} a^{15} - \frac{32320779092807940653}{6605656525298614423257} a^{14} + \frac{553486963432542815821}{6605656525298614423257} a^{13} - \frac{751635519557043715978}{6605656525298614423257} a^{12} + \frac{45976896807696769700}{733961836144290491473} a^{11} - \frac{23294629751823249580}{153619919192991033099} a^{10} + \frac{798358765025265938339}{6605656525298614423257} a^{9} - \frac{2362957697401184501797}{6605656525298614423257} a^{8} + \frac{1952719713862510691992}{6605656525298614423257} a^{7} + \frac{894800964405426372638}{2201885508432871474419} a^{6} - \frac{75786439088036401255}{733961836144290491473} a^{5} - \frac{1069472200868886021736}{6605656525298614423257} a^{4} + \frac{722334231097057005836}{2201885508432871474419} a^{3} - \frac{9230979092860655618}{19485712464007712163} a^{2} + \frac{2769954635502334768382}{6605656525298614423257} a - \frac{34035947387956052071}{111960280089807024123}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( -\frac{327794166010}{7077992177263971} a^{17} + \frac{10077315797}{39988656368723} a^{16} - \frac{3323881744883}{7077992177263971} a^{15} + \frac{19571347987856}{7077992177263971} a^{14} - \frac{10995799911761}{2359330725754657} a^{13} + \frac{104803737556591}{7077992177263971} a^{12} - \frac{232197251786639}{7077992177263971} a^{11} + \frac{601291622928835}{7077992177263971} a^{10} - \frac{346554137295767}{2359330725754657} a^{9} + \frac{2749488962142335}{7077992177263971} a^{8} - \frac{3979145666377667}{7077992177263971} a^{7} + \frac{7458633676328408}{7077992177263971} a^{6} - \frac{3814589994352252}{2359330725754657} a^{5} + \frac{4630859569905738}{2359330725754657} a^{4} - \frac{5145949107069858}{2359330725754657} a^{3} + \frac{12891799461895493}{7077992177263971} a^{2} - \frac{7943362345517017}{7077992177263971} a + \frac{44820809411551}{39988656368723} \) (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:  \( 1439143.7292234616 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3^2:S_3$ (as 18T24):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 10 conjugacy class representatives for $C_3^2:S_3$
Character table for $C_3^2:S_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.108.1 x3, 6.0.34992.1, 9.3.4306704645312.3

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

Sibling fields

Degree 9 siblings: data not computed
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 R R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
2Data not computed
3Data not computed
$43$43.3.2.3$x^{3} - 3483$$3$$1$$2$$C_3$$[\ ]_{3}$
43.3.0.1$x^{3} - x + 10$$1$$3$$0$$C_3$$[\ ]^{3}$
43.3.2.2$x^{3} + 387$$3$$1$$2$$C_3$$[\ ]_{3}$
43.3.2.2$x^{3} + 387$$3$$1$$2$$C_3$$[\ ]_{3}$
43.3.2.3$x^{3} - 3483$$3$$1$$2$$C_3$$[\ ]_{3}$
43.3.0.1$x^{3} - x + 10$$1$$3$$0$$C_3$$[\ ]^{3}$