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Properties

Label 18.0.91176404162...0000.4
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{24}\cdot 5^{9}\cdot 7^{9}$
Root discriminant $40.63$
Ramified primes $2, 3, 5, 7$
Class number $54$ (GRH)
Class group $[3, 18]$ (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![122500, 147000, 182700, 273560, 188904, 63342, 67349, 40986, 10161, 2976, -972, 198, 371, 162, 72, -4, 9, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 9*x^16 - 4*x^15 + 72*x^14 + 162*x^13 + 371*x^12 + 198*x^11 - 972*x^10 + 2976*x^9 + 10161*x^8 + 40986*x^7 + 67349*x^6 + 63342*x^5 + 188904*x^4 + 273560*x^3 + 182700*x^2 + 147000*x + 122500)
 
gp: K = bnfinit(x^18 + 9*x^16 - 4*x^15 + 72*x^14 + 162*x^13 + 371*x^12 + 198*x^11 - 972*x^10 + 2976*x^9 + 10161*x^8 + 40986*x^7 + 67349*x^6 + 63342*x^5 + 188904*x^4 + 273560*x^3 + 182700*x^2 + 147000*x + 122500, 1)
 

Normalized defining polynomial

\( x^{18} + 9 x^{16} - 4 x^{15} + 72 x^{14} + 162 x^{13} + 371 x^{12} + 198 x^{11} - 972 x^{10} + 2976 x^{9} + 10161 x^{8} + 40986 x^{7} + 67349 x^{6} + 63342 x^{5} + 188904 x^{4} + 273560 x^{3} + 182700 x^{2} + 147000 x + 122500 \)

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:  \(-91176404162771495736000000000=-\,2^{12}\cdot 3^{24}\cdot 5^{9}\cdot 7^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40.63$
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:  $2, 3, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{10} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{5} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{2}{5} a$, $\frac{1}{70} a^{10} + \frac{1}{70} a^{9} + \frac{2}{7} a^{8} + \frac{3}{14} a^{7} - \frac{1}{35} a^{6} - \frac{17}{70} a^{5} + \frac{1}{14} a^{4} - \frac{2}{7} a^{3} - \frac{12}{35} a^{2} - \frac{1}{5} a$, $\frac{1}{70} a^{11} - \frac{1}{35} a^{9} + \frac{3}{7} a^{8} - \frac{17}{70} a^{7} + \frac{2}{7} a^{6} - \frac{3}{35} a^{5} + \frac{1}{7} a^{4} + \frac{31}{70} a^{3} + \frac{1}{7} a^{2} + \frac{2}{5} a$, $\frac{1}{70} a^{12} - \frac{3}{70} a^{9} - \frac{6}{35} a^{8} - \frac{2}{7} a^{7} + \frac{5}{14} a^{6} - \frac{12}{35} a^{5} + \frac{3}{35} a^{4} + \frac{1}{14} a^{3} - \frac{2}{7} a^{2} - \frac{2}{5} a$, $\frac{1}{70} a^{13} - \frac{1}{35} a^{9} + \frac{1}{14} a^{8} + \frac{1}{14} a^{6} + \frac{11}{70} a^{5} - \frac{3}{14} a^{4} + \frac{5}{14} a^{3} - \frac{3}{7} a^{2}$, $\frac{1}{700} a^{14} + \frac{3}{700} a^{13} - \frac{3}{700} a^{12} + \frac{1}{175} a^{11} - \frac{3}{700} a^{10} - \frac{1}{700} a^{9} - \frac{339}{700} a^{8} - \frac{22}{175} a^{7} + \frac{33}{700} a^{6} + \frac{13}{700} a^{5} - \frac{3}{700} a^{4} - \frac{53}{350} a^{3} + \frac{61}{175} a^{2} - \frac{1}{10} a$, $\frac{1}{3500} a^{15} - \frac{1}{1750} a^{14} - \frac{2}{875} a^{13} + \frac{19}{3500} a^{12} - \frac{3}{3500} a^{11} + \frac{6}{875} a^{10} - \frac{26}{875} a^{9} - \frac{149}{500} a^{8} + \frac{283}{3500} a^{7} - \frac{561}{1750} a^{6} - \frac{377}{875} a^{5} + \frac{9}{3500} a^{4} + \frac{361}{875} a^{3} - \frac{23}{350} a^{2} - \frac{1}{10} a$, $\frac{1}{49808500} a^{16} - \frac{363}{9961700} a^{15} - \frac{32657}{49808500} a^{14} - \frac{45401}{24904250} a^{13} + \frac{1331}{284620} a^{12} + \frac{47613}{49808500} a^{11} + \frac{224009}{49808500} a^{10} - \frac{677583}{24904250} a^{9} - \frac{19265133}{49808500} a^{8} - \frac{201393}{7115500} a^{7} + \frac{16101753}{49808500} a^{6} - \frac{1872331}{24904250} a^{5} + \frac{2857313}{12452125} a^{4} + \frac{41663}{254125} a^{3} - \frac{1442}{50825} a^{2} - \frac{2726}{10165} a - \frac{9}{2033}$, $\frac{1}{1266157908687432097169780500} a^{17} + \frac{1595589063889120771}{1266157908687432097169780500} a^{16} - \frac{42994440218103722330373}{316539477171858024292445125} a^{15} + \frac{575166109069646308058041}{1266157908687432097169780500} a^{14} - \frac{1703005985989842417674768}{316539477171858024292445125} a^{13} + \frac{3173734856230026058988443}{1266157908687432097169780500} a^{12} - \frac{1673110947447090159894792}{316539477171858024292445125} a^{11} + \frac{720472927102090607334909}{180879701241061728167111500} a^{10} + \frac{5869971339150645038162054}{316539477171858024292445125} a^{9} - \frac{200495036272274944053064249}{1266157908687432097169780500} a^{8} + \frac{315247882186872982879130471}{633078954343716048584890250} a^{7} + \frac{1667932915732757402635293}{180879701241061728167111500} a^{6} + \frac{1352089881744429136715811}{13327977986183495759681900} a^{5} - \frac{258052614539342161341056}{666398899309174787984095} a^{4} - \frac{16293687160963190597940453}{90439850620530864083555750} a^{3} - \frac{8466374072534420873854707}{18087970124106172816711150} a^{2} + \frac{100122686198350292545776}{258399573201516754524445} a - \frac{7678945245728718662529}{51679914640303350904889}$

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

Class group and class number

$C_{3}\times C_{18}$, which has order $54$ (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:  \( 5817988.03207 \) (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

\(\Q(\sqrt{-35}) \), 3.1.140.1 x3, 3.1.11340.1 x3, 3.1.2835.1 x3, 3.1.11340.2 x3, 6.0.686000.1, 6.0.4500846000.2, 6.0.281302875.1, 6.0.4500846000.1, 9.1.51039593640000.10 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 R R R R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
$2$2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$3$3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
$5$5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
$7$7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$