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Properties

Label 18.0.14026877608...1968.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{21}\cdot 41^{9}$
Root discriminant $36.62$
Ramified primes $2, 3, 41$
Class number $24$ (GRH)
Class group $[2, 2, 6]$ (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![123, 0, -3612, 0, 33456, 0, -6005, 0, 5412, 0, -312, 0, 205, 0, 24, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 24*x^14 + 205*x^12 - 312*x^10 + 5412*x^8 - 6005*x^6 + 33456*x^4 - 3612*x^2 + 123)
 
gp: K = bnfinit(x^18 + 24*x^14 + 205*x^12 - 312*x^10 + 5412*x^8 - 6005*x^6 + 33456*x^4 - 3612*x^2 + 123, 1)
 

Normalized defining polynomial

\( x^{18} + 24 x^{14} + 205 x^{12} - 312 x^{10} + 5412 x^{8} - 6005 x^{6} + 33456 x^{4} - 3612 x^{2} + 123 \)

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:  \(-14026877608108874998608211968=-\,2^{12}\cdot 3^{21}\cdot 41^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.62$
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:  $2, 3, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{14} a^{10} - \frac{2}{7} a^{8} - \frac{1}{2} a^{7} - \frac{3}{14} a^{4} + \frac{3}{7} a^{2} - \frac{1}{2} a - \frac{3}{7}$, $\frac{1}{14} a^{11} + \frac{3}{14} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{3}{14} a^{5} - \frac{1}{14} a^{3} - \frac{1}{2} a^{2} - \frac{3}{7} a - \frac{1}{2}$, $\frac{1}{140} a^{12} + \frac{1}{70} a^{10} + \frac{8}{35} a^{8} - \frac{13}{35} a^{6} - \frac{13}{70} a^{4} - \frac{1}{2} a^{3} + \frac{11}{35} a^{2} - \frac{43}{140}$, $\frac{1}{140} a^{13} + \frac{1}{70} a^{11} + \frac{8}{35} a^{9} - \frac{13}{35} a^{7} - \frac{13}{70} a^{5} - \frac{1}{2} a^{4} + \frac{11}{35} a^{3} - \frac{43}{140} a$, $\frac{1}{140} a^{14} - \frac{1}{70} a^{10} + \frac{1}{35} a^{8} - \frac{1}{2} a^{7} - \frac{31}{70} a^{6} - \frac{1}{2} a^{5} + \frac{23}{70} a^{4} - \frac{31}{140} a^{2} - \frac{1}{2} a - \frac{1}{10}$, $\frac{1}{140} a^{15} - \frac{1}{70} a^{11} + \frac{1}{35} a^{9} - \frac{1}{2} a^{8} - \frac{31}{70} a^{7} - \frac{1}{2} a^{6} + \frac{23}{70} a^{5} - \frac{31}{140} a^{3} - \frac{1}{2} a^{2} - \frac{1}{10} a$, $\frac{1}{226327004540} a^{16} - \frac{353087821}{226327004540} a^{14} - \frac{4022342}{1616621461} a^{12} - \frac{416255431}{22632700454} a^{10} + \frac{363638789}{113163502270} a^{8} - \frac{34827165193}{113163502270} a^{6} - \frac{1}{2} a^{5} + \frac{85298772631}{226327004540} a^{4} + \frac{824336069}{45265400908} a^{2} - \frac{1}{2} a - \frac{3337061531}{113163502270}$, $\frac{1}{226327004540} a^{17} - \frac{353087821}{226327004540} a^{15} - \frac{4022342}{1616621461} a^{13} - \frac{416255431}{22632700454} a^{11} + \frac{363638789}{113163502270} a^{9} - \frac{34827165193}{113163502270} a^{7} - \frac{1}{2} a^{6} + \frac{85298772631}{226327004540} a^{5} + \frac{824336069}{45265400908} a^{3} - \frac{1}{2} a^{2} - \frac{3337061531}{113163502270} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{6}$, which has order $24$ (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:  \( 1315359.70248 \) (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

\(\Q(\sqrt{-123}) \), 3.1.4428.2 x3, 3.1.492.1 x3, 3.1.1107.1 x3, 3.1.4428.1 x3, 6.0.2411683632.2, 6.0.29773872.1, 6.0.150730227.1, 6.0.2411683632.1, 9.1.10678935122496.3 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 ${\href{/LocalNumberField/5.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$$[\ ]_{3}^{2}$
$3$3.6.7.1$x^{6} + 6 x^{2} + 6$$6$$1$$7$$S_3$$[3/2]_{2}$
3.6.7.1$x^{6} + 6 x^{2} + 6$$6$$1$$7$$S_3$$[3/2]_{2}$
3.6.7.1$x^{6} + 6 x^{2} + 6$$6$$1$$7$$S_3$$[3/2]_{2}$
$41$41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$