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Properties

Label 18.0.18572676423...1851.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,11^{9}\cdot 31^{12}$
Root discriminant $32.73$
Ramified primes $11, 31$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $S_3 \times C_3$ (as 18T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5324, 0, 16577, 0, -23210, 0, 6301, 0, 2410, 0, -780, 0, 278, 0, 21, 0, -10, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 10*x^16 + 21*x^14 + 278*x^12 - 780*x^10 + 2410*x^8 + 6301*x^6 - 23210*x^4 + 16577*x^2 + 5324)
 
gp: K = bnfinit(x^18 - 10*x^16 + 21*x^14 + 278*x^12 - 780*x^10 + 2410*x^8 + 6301*x^6 - 23210*x^4 + 16577*x^2 + 5324, 1)
 

Normalized defining polynomial

\( x^{18} - 10 x^{16} + 21 x^{14} + 278 x^{12} - 780 x^{10} + 2410 x^{8} + 6301 x^{6} - 23210 x^{4} + 16577 x^{2} + 5324 \)

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:  \(-1857267642320843141188551851=-\,11^{9}\cdot 31^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.73$
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:  $11, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{7} - \frac{1}{2} a^{3} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{12} a^{8} - \frac{5}{12} a^{2} + \frac{1}{3}$, $\frac{1}{12} a^{9} - \frac{5}{12} a^{3} + \frac{1}{3} a$, $\frac{1}{24} a^{10} - \frac{1}{24} a^{9} - \frac{1}{24} a^{8} - \frac{5}{24} a^{4} - \frac{7}{24} a^{3} - \frac{1}{8} a^{2} + \frac{1}{3} a - \frac{1}{6}$, $\frac{1}{24} a^{11} - \frac{1}{24} a^{8} - \frac{5}{24} a^{5} + \frac{1}{6} a^{3} + \frac{5}{24} a^{2} - \frac{1}{6}$, $\frac{1}{24} a^{12} - \frac{1}{24} a^{9} - \frac{5}{24} a^{6} + \frac{1}{6} a^{4} + \frac{5}{24} a^{3} - \frac{1}{6} a$, $\frac{1}{48} a^{13} - \frac{1}{48} a^{12} - \frac{1}{48} a^{11} - \frac{1}{48} a^{10} + \frac{1}{48} a^{9} + \frac{1}{48} a^{8} - \frac{5}{48} a^{7} - \frac{7}{48} a^{6} - \frac{1}{16} a^{5} - \frac{11}{48} a^{4} - \frac{7}{16} a^{3} + \frac{1}{16} a^{2} + \frac{1}{3} a - \frac{5}{12}$, $\frac{1}{2112} a^{14} - \frac{1}{66} a^{12} - \frac{1}{176} a^{10} - \frac{1}{24} a^{9} - \frac{37}{1056} a^{8} + \frac{113}{528} a^{6} - \frac{41}{528} a^{4} - \frac{7}{24} a^{3} - \frac{173}{704} a^{2} - \frac{1}{6} a + \frac{3}{16}$, $\frac{1}{4224} a^{15} - \frac{1}{4224} a^{14} - \frac{1}{132} a^{13} + \frac{1}{132} a^{12} + \frac{19}{1056} a^{11} - \frac{19}{1056} a^{10} - \frac{27}{704} a^{9} - \frac{7}{2112} a^{8} + \frac{113}{1056} a^{7} - \frac{113}{1056} a^{6} + \frac{113}{1056} a^{5} - \frac{113}{1056} a^{4} + \frac{443}{1408} a^{3} + \frac{1663}{4224} a^{2} - \frac{47}{96} a + \frac{31}{96}$, $\frac{1}{4815178435584} a^{16} - \frac{5782927}{1605059478528} a^{14} + \frac{871568727}{133754956544} a^{12} + \frac{8550967729}{2407589217792} a^{10} - \frac{1}{24} a^{9} + \frac{45138747847}{2407589217792} a^{8} + \frac{1038517369}{12539527176} a^{6} + \frac{423096045469}{4815178435584} a^{4} + \frac{5}{24} a^{3} - \frac{17402996867}{437743494144} a^{2} + \frac{1}{3} a - \frac{2477531399}{9948715776}$, $\frac{1}{9630356871168} a^{17} - \frac{1}{9630356871168} a^{16} - \frac{5782927}{3210118957056} a^{15} + \frac{5782927}{3210118957056} a^{14} + \frac{871568727}{267509913088} a^{13} - \frac{871568727}{267509913088} a^{12} - \frac{91765249679}{4815178435584} a^{11} + \frac{91765249679}{4815178435584} a^{10} - \frac{55177469561}{4815178435584} a^{9} + \frac{55177469561}{4815178435584} a^{8} + \frac{1038517369}{25079054352} a^{7} + \frac{5231246219}{25079054352} a^{6} + \frac{1426258219549}{9630356871168} a^{5} - \frac{1426258219549}{9630356871168} a^{4} - \frac{218035431683}{875486988288} a^{3} - \frac{219708062461}{875486988288} a^{2} + \frac{838707193}{19897431552} a + \frac{4135650695}{19897431552}$

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:  $8$
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:  \( 10795929.0484 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3$ (as 18T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 9 conjugacy class representatives for $S_3 \times C_3$
Character table for $S_3 \times C_3$

Intermediate fields

\(\Q(\sqrt{-11}) \), 3.1.10571.1 x3, 3.3.961.1, 6.0.1229206451.1, 6.0.1279091.2 x2, 6.0.1229206451.2, 9.3.1181267399411.1 x3

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

Sibling fields

Degree 6 sibling: 6.0.1279091.2
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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$11$11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$31$31.9.6.1$x^{9} + 837 x^{6} + 232562 x^{3} + 21717639$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
31.9.6.1$x^{9} + 837 x^{6} + 232562 x^{3} + 21717639$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$