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Properties

Label 18.0.72656651259...2352.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{9}\cdot 37^{14}$
Root discriminant $45.60$
Ramified primes $2, 3, 37$
Class number $108$ (GRH)
Class group $[3, 6, 6]$ (GRH)
Galois group $S_3 \times C_6$ (as 18T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 10, 69, 264, 766, 1369, 2058, 1956, 2434, 1863, 1779, 814, 565, 149, 123, 17, 14, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 + 14*x^16 + 17*x^15 + 123*x^14 + 149*x^13 + 565*x^12 + 814*x^11 + 1779*x^10 + 1863*x^9 + 2434*x^8 + 1956*x^7 + 2058*x^6 + 1369*x^5 + 766*x^4 + 264*x^3 + 69*x^2 + 10*x + 1)
 
gp: K = bnfinit(x^18 - x^17 + 14*x^16 + 17*x^15 + 123*x^14 + 149*x^13 + 565*x^12 + 814*x^11 + 1779*x^10 + 1863*x^9 + 2434*x^8 + 1956*x^7 + 2058*x^6 + 1369*x^5 + 766*x^4 + 264*x^3 + 69*x^2 + 10*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} + 14 x^{16} + 17 x^{15} + 123 x^{14} + 149 x^{13} + 565 x^{12} + 814 x^{11} + 1779 x^{10} + 1863 x^{9} + 2434 x^{8} + 1956 x^{7} + 2058 x^{6} + 1369 x^{5} + 766 x^{4} + 264 x^{3} + 69 x^{2} + 10 x + 1 \)

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:  \(-726566512595229689293941092352=-\,2^{12}\cdot 3^{9}\cdot 37^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.60$
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:  $2, 3, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{65749} a^{15} - \frac{5332}{65749} a^{14} + \frac{16007}{65749} a^{13} + \frac{25341}{65749} a^{12} - \frac{25201}{65749} a^{11} + \frac{24163}{65749} a^{10} + \frac{25809}{65749} a^{9} - \frac{25962}{65749} a^{8} + \frac{11634}{65749} a^{7} + \frac{13710}{65749} a^{6} + \frac{2314}{65749} a^{5} + \frac{26758}{65749} a^{4} + \frac{14500}{65749} a^{3} - \frac{9686}{65749} a^{2} - \frac{24376}{65749} a + \frac{30970}{65749}$, $\frac{1}{65749} a^{16} - \frac{10649}{65749} a^{14} + \frac{32463}{65749} a^{13} - \frac{21184}{65749} a^{12} - \frac{22362}{65749} a^{11} - \frac{5115}{65749} a^{10} - \frac{25031}{65749} a^{9} - \frac{16105}{65749} a^{8} - \frac{20858}{65749} a^{7} - \frac{8854}{65749} a^{6} + \frac{4194}{65749} a^{5} + \frac{12826}{65749} a^{4} - \frac{16510}{65749} a^{3} + \frac{8586}{65749} a^{2} - \frac{21838}{65749} a - \frac{29448}{65749}$, $\frac{1}{7361323789} a^{17} + \frac{50244}{7361323789} a^{16} + \frac{13166}{7361323789} a^{15} - \frac{485066914}{7361323789} a^{14} - \frac{2662234411}{7361323789} a^{13} - \frac{21332628}{71469163} a^{12} + \frac{2032254116}{7361323789} a^{11} + \frac{2571505486}{7361323789} a^{10} + \frac{3282051468}{7361323789} a^{9} - \frac{514626392}{7361323789} a^{8} + \frac{1998018890}{7361323789} a^{7} - \frac{3612714635}{7361323789} a^{6} + \frac{1592330147}{7361323789} a^{5} - \frac{1532665392}{7361323789} a^{4} + \frac{2777868532}{7361323789} a^{3} - \frac{7710804}{71469163} a^{2} + \frac{1942367580}{7361323789} a - \frac{2773415866}{7361323789}$

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

Class group and class number

$C_{3}\times C_{6}\times C_{6}$, which has order $108$ (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:  \( -\frac{1350662074}{7361323789} a^{17} + \frac{1551632519}{7361323789} a^{16} - \frac{19168353112}{7361323789} a^{15} - \frac{20077037170}{7361323789} a^{14} - \frac{163553918972}{7361323789} a^{13} - \frac{1721605119}{71469163} a^{12} - \frac{740295754360}{7361323789} a^{11} - \frac{26842502770}{198954697} a^{10} - \frac{2271696792466}{7361323789} a^{9} - \frac{2200311098785}{7361323789} a^{8} - \frac{3012608895078}{7361323789} a^{7} - \frac{2246037552002}{7361323789} a^{6} - \frac{2522699684024}{7361323789} a^{5} - \frac{1531437507749}{7361323789} a^{4} - \frac{872110497300}{7361323789} a^{3} - \frac{2532268388}{71469163} a^{2} - \frac{80240135046}{7361323789} a - \frac{4330194133}{7361323789} \) (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:  \( 615797.1340659427 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6\times S_3$ (as 18T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 18 conjugacy class representatives for $S_3 \times C_6$
Character table for $S_3 \times C_6$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.3.148.1, 3.3.1369.1, 6.0.591408.1, 6.0.50602347.1, 9.9.6075640136512.1

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

Sibling fields

Galois closure: data not computed
Degree 12 sibling: data not computed
Degree 18 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
2Data not computed
$3$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$37$37.3.2.1$x^{3} - 37$$3$$1$$2$$C_3$$[\ ]_{3}$
37.3.2.1$x^{3} - 37$$3$$1$$2$$C_3$$[\ ]_{3}$
37.6.5.1$x^{6} - 37$$6$$1$$5$$C_6$$[\ ]_{6}$
37.6.5.1$x^{6} - 37$$6$$1$$5$$C_6$$[\ ]_{6}$