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Properties

Label 18.18.1164070036...0000.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{12}\cdot 3^{24}\cdot 5^{9}\cdot 61^{6}$
Root discriminant $60.46$
Ramified primes $2, 3, 5, 61$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3\times C_3:S_3$ (as 18T23)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![125, 1500, 4950, -2525, -34320, -18900, 84016, 52290, -104526, -39545, 65196, 7470, -16668, -120, 1692, -15, -72, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 72*x^16 - 15*x^15 + 1692*x^14 - 120*x^13 - 16668*x^12 + 7470*x^11 + 65196*x^10 - 39545*x^9 - 104526*x^8 + 52290*x^7 + 84016*x^6 - 18900*x^5 - 34320*x^4 - 2525*x^3 + 4950*x^2 + 1500*x + 125)
 
gp: K = bnfinit(x^18 - 72*x^16 - 15*x^15 + 1692*x^14 - 120*x^13 - 16668*x^12 + 7470*x^11 + 65196*x^10 - 39545*x^9 - 104526*x^8 + 52290*x^7 + 84016*x^6 - 18900*x^5 - 34320*x^4 - 2525*x^3 + 4950*x^2 + 1500*x + 125, 1)
 

Normalized defining polynomial

\( x^{18} - 72 x^{16} - 15 x^{15} + 1692 x^{14} - 120 x^{13} - 16668 x^{12} + 7470 x^{11} + 65196 x^{10} - 39545 x^{9} - 104526 x^{8} + 52290 x^{7} + 84016 x^{6} - 18900 x^{5} - 34320 x^{4} - 2525 x^{3} + 4950 x^{2} + 1500 x + 125 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(116407003600838612509128000000000=2^{12}\cdot 3^{24}\cdot 5^{9}\cdot 61^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $60.46$
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:  $2, 3, 5, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{5} a^{14} - \frac{2}{5} a^{12} + \frac{2}{5} a^{10} + \frac{2}{5} a^{8} + \frac{1}{5} a^{6} - \frac{1}{5} a^{4} + \frac{1}{5} a^{2}$, $\frac{1}{135} a^{15} - \frac{14}{45} a^{13} - \frac{13}{27} a^{12} + \frac{1}{5} a^{11} + \frac{2}{9} a^{10} - \frac{53}{135} a^{9} + \frac{2}{45} a^{7} - \frac{8}{27} a^{6} + \frac{1}{15} a^{5} + \frac{2}{9} a^{4} + \frac{7}{45} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{13}{27}$, $\frac{1}{675} a^{16} - \frac{14}{225} a^{14} - \frac{13}{135} a^{13} - \frac{9}{25} a^{12} + \frac{2}{45} a^{11} + \frac{217}{675} a^{10} + \frac{2}{5} a^{9} + \frac{2}{225} a^{8} + \frac{46}{135} a^{7} + \frac{31}{75} a^{6} + \frac{11}{45} a^{5} - \frac{38}{225} a^{4} - \frac{1}{3} a^{3} - \frac{2}{15} a^{2} - \frac{8}{27} a$, $\frac{1}{8666240304154206862258102575} a^{17} + \frac{1599983118815446554688162}{8666240304154206862258102575} a^{16} - \frac{16512688115586755252802182}{8666240304154206862258102575} a^{15} + \frac{81574929359997882560166691}{8666240304154206862258102575} a^{14} - \frac{1903248678321290372796629918}{8666240304154206862258102575} a^{13} - \frac{1518336231191910812784628256}{8666240304154206862258102575} a^{12} - \frac{47740717686069351022368128}{8666240304154206862258102575} a^{11} - \frac{2301901162619861474685732956}{8666240304154206862258102575} a^{10} + \frac{389092826355825893834238841}{8666240304154206862258102575} a^{9} + \frac{3918600620489319735540245897}{8666240304154206862258102575} a^{8} + \frac{524250042387149898361546274}{8666240304154206862258102575} a^{7} + \frac{1221436113828577432139257448}{8666240304154206862258102575} a^{6} - \frac{1373688503345357711002301623}{2888746768051402287419367525} a^{5} - \frac{242980551423539135839267426}{2888746768051402287419367525} a^{4} + \frac{253558388056499078157760913}{577749353610280457483873505} a^{3} + \frac{181335923706291826291591856}{1733248060830841372451620515} a^{2} + \frac{54454827877247755941969439}{346649612166168274490324103} a - \frac{165345776080769777311795262}{346649612166168274490324103}$

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:  $17$
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:  \( 13504431477.0 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_3:S_3$ (as 18T23):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 18 conjugacy class representatives for $C_3\times C_3:S_3$
Character table for $C_3\times C_3:S_3$

Intermediate fields

\(\Q(\sqrt{5}) \), 3.3.1620.1 x3, 6.6.7442000.1, 6.6.48826962000.4, 6.6.3051685125.2, 6.6.13122000.1

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 R R R ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
2Data not computed
$3$3.6.8.5$x^{6} + 9 x^{2} + 9$$3$$2$$8$$S_3$$[2]^{2}$
3.6.8.5$x^{6} + 9 x^{2} + 9$$3$$2$$8$$S_3$$[2]^{2}$
3.6.8.5$x^{6} + 9 x^{2} + 9$$3$$2$$8$$S_3$$[2]^{2}$
$5$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$61$61.3.2.3$x^{3} - 244$$3$$1$$2$$C_3$$[\ ]_{3}$
61.3.2.3$x^{3} - 244$$3$$1$$2$$C_3$$[\ ]_{3}$
61.3.0.1$x^{3} - x + 10$$1$$3$$0$$C_3$$[\ ]^{3}$
61.3.2.3$x^{3} - 244$$3$$1$$2$$C_3$$[\ ]_{3}$
61.3.0.1$x^{3} - x + 10$$1$$3$$0$$C_3$$[\ ]^{3}$
61.3.0.1$x^{3} - x + 10$$1$$3$$0$$C_3$$[\ ]^{3}$