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Properties

Label 18.0.13710594150...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{26}\cdot 3^{21}\cdot 5^{9}$
Root discriminant $21.93$
Ramified primes $2, 3, 5$
Class number $2$
Class group $[2]$
Galois group $C_2\times S_3^2$ (as 18T29)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, -24, 96, -266, 570, -954, 1180, -918, 180, 505, -651, 378, -76, -84, 120, -84, 36, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 36*x^16 - 84*x^15 + 120*x^14 - 84*x^13 - 76*x^12 + 378*x^11 - 651*x^10 + 505*x^9 + 180*x^8 - 918*x^7 + 1180*x^6 - 954*x^5 + 570*x^4 - 266*x^3 + 96*x^2 - 24*x + 4)
 
gp: K = bnfinit(x^18 - 9*x^17 + 36*x^16 - 84*x^15 + 120*x^14 - 84*x^13 - 76*x^12 + 378*x^11 - 651*x^10 + 505*x^9 + 180*x^8 - 918*x^7 + 1180*x^6 - 954*x^5 + 570*x^4 - 266*x^3 + 96*x^2 - 24*x + 4, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 36 x^{16} - 84 x^{15} + 120 x^{14} - 84 x^{13} - 76 x^{12} + 378 x^{11} - 651 x^{10} + 505 x^{9} + 180 x^{8} - 918 x^{7} + 1180 x^{6} - 954 x^{5} + 570 x^{4} - 266 x^{3} + 96 x^{2} - 24 x + 4 \)

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:  \(-1371059415023616000000000=-\,2^{26}\cdot 3^{21}\cdot 5^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.93$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6}$, $\frac{1}{6} a^{13} + \frac{1}{6} a^{12} - \frac{1}{2} a^{11} + \frac{1}{6} a^{10} + \frac{1}{6} a^{9} - \frac{1}{2} a^{8} - \frac{1}{6} a^{7} - \frac{1}{6} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{90} a^{14} - \frac{7}{90} a^{13} - \frac{1}{45} a^{12} + \frac{13}{90} a^{11} + \frac{1}{45} a^{10} - \frac{41}{90} a^{9} + \frac{2}{9} a^{8} - \frac{7}{18} a^{7} - \frac{1}{90} a^{6} + \frac{7}{45} a^{5} + \frac{11}{45} a^{4} + \frac{16}{45} a^{3} + \frac{17}{45} a^{2} + \frac{19}{45} a - \frac{13}{45}$, $\frac{1}{270} a^{15} - \frac{1}{45} a^{13} - \frac{1}{270} a^{12} - \frac{7}{45} a^{11} - \frac{1}{10} a^{10} - \frac{7}{45} a^{9} + \frac{7}{18} a^{8} - \frac{7}{90} a^{7} + \frac{7}{270} a^{6} + \frac{4}{9} a^{5} + \frac{16}{45} a^{4} - \frac{2}{45} a^{3} + \frac{1}{45} a^{2} - \frac{1}{9} a - \frac{1}{135}$, $\frac{1}{20250} a^{16} - \frac{4}{10125} a^{15} - \frac{17}{6750} a^{14} + \frac{497}{20250} a^{13} + \frac{2323}{10125} a^{12} + \frac{1933}{6750} a^{11} - \frac{1156}{3375} a^{10} + \frac{119}{2250} a^{9} - \frac{2837}{6750} a^{8} + \frac{1727}{4050} a^{7} + \frac{2809}{20250} a^{6} + \frac{457}{1125} a^{5} + \frac{4}{675} a^{4} - \frac{448}{3375} a^{3} + \frac{1442}{3375} a^{2} - \frac{1006}{10125} a + \frac{1538}{10125}$, $\frac{1}{60750} a^{17} - \frac{1}{60750} a^{16} - \frac{107}{60750} a^{15} + \frac{14}{6075} a^{14} - \frac{8}{243} a^{13} - \frac{6152}{30375} a^{12} + \frac{3922}{10125} a^{11} + \frac{524}{10125} a^{10} + \frac{9787}{20250} a^{9} + \frac{19933}{60750} a^{8} + \frac{12629}{60750} a^{7} + \frac{8882}{30375} a^{6} + \frac{2867}{10125} a^{5} - \frac{3683}{10125} a^{4} + \frac{1681}{10125} a^{3} - \frac{1099}{30375} a^{2} - \frac{5504}{30375} a - \frac{9484}{30375}$

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

Class group and class number

$C_{2}$, which has order $2$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 398664.7871956061 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times S_3^2$ (as 18T29):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 72
The 18 conjugacy class representatives for $C_2\times S_3^2$
Character table for $C_2\times S_3^2$

Intermediate fields

\(\Q(\sqrt{-15}) \), 3.1.108.1, 3.1.1080.1, 6.0.4374000.1, 6.0.17496000.1, 9.1.60466176000.1

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

Sibling fields

Degree 12 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 R ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.6.11.1$x^{6} + 14$$6$$1$$11$$D_{6}$$[3]_{3}^{2}$
2.6.11.1$x^{6} + 14$$6$$1$$11$$D_{6}$$[3]_{3}^{2}$
$3$3.6.7.2$x^{6} + 3 x^{2} + 6$$6$$1$$7$$D_{6}$$[3/2]_{2}^{2}$
3.12.14.6$x^{12} + 3 x^{11} + 3 x^{10} - 6 x^{9} + 3 x^{8} + 9 x^{7} + 9 x^{4} + 9 x^{3} + 9$$6$$2$$14$$D_6$$[3/2]_{2}^{2}$
$5$5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$