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Properties

Label 18.0.99221394673...3472.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 307^{9}$
Root discriminant $27.81$
Ramified primes $2, 307$
Class number $9$
Class group $[3, 3]$
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![1228, 0, 400, 0, -576, 0, 1929, 0, 610, 0, 103, 0, -108, 0, 15, 0, -2, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^16 + 15*x^14 - 108*x^12 + 103*x^10 + 610*x^8 + 1929*x^6 - 576*x^4 + 400*x^2 + 1228)
 
gp: K = bnfinit(x^18 - 2*x^16 + 15*x^14 - 108*x^12 + 103*x^10 + 610*x^8 + 1929*x^6 - 576*x^4 + 400*x^2 + 1228, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{16} + 15 x^{14} - 108 x^{12} + 103 x^{10} + 610 x^{8} + 1929 x^{6} - 576 x^{4} + 400 x^{2} + 1228 \)

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:  \(-99221394673320004142313472=-\,2^{12}\cdot 307^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.81$
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:  $2, 307$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{7} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{12} a^{8} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{3}$, $\frac{1}{12} a^{9} + \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{3} a$, $\frac{1}{12} a^{10} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{3} a^{2}$, $\frac{1}{12} a^{11} - \frac{1}{2} a^{5} + \frac{5}{12} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{96} a^{12} - \frac{1}{32} a^{10} + \frac{1}{32} a^{8} + \frac{1}{32} a^{6} + \frac{1}{12} a^{4} - \frac{1}{2} a^{3} - \frac{3}{8} a^{2} + \frac{3}{8}$, $\frac{1}{96} a^{13} - \frac{1}{32} a^{11} + \frac{1}{32} a^{9} + \frac{1}{32} a^{7} + \frac{1}{12} a^{5} - \frac{1}{2} a^{4} - \frac{3}{8} a^{3} + \frac{3}{8} a$, $\frac{1}{960} a^{14} + \frac{1}{480} a^{12} + \frac{1}{48} a^{10} - \frac{7}{480} a^{8} + \frac{167}{960} a^{6} - \frac{1}{2} a^{5} - \frac{7}{48} a^{4} - \frac{17}{60} a^{2} - \frac{1}{2} a + \frac{29}{240}$, $\frac{1}{1920} a^{15} - \frac{1}{1920} a^{14} - \frac{1}{240} a^{13} - \frac{1}{960} a^{12} + \frac{5}{192} a^{11} + \frac{1}{32} a^{10} + \frac{3}{160} a^{9} - \frac{11}{320} a^{8} + \frac{137}{1920} a^{7} + \frac{73}{1920} a^{6} + \frac{25}{96} a^{5} + \frac{43}{96} a^{4} + \frac{11}{240} a^{3} - \frac{1}{40} a^{2} + \frac{33}{160} a + \frac{17}{160}$, $\frac{1}{4209747840} a^{16} + \frac{28637}{467749760} a^{14} - \frac{1}{192} a^{13} + \frac{407261}{350812320} a^{12} - \frac{5}{192} a^{11} + \frac{881853}{58468720} a^{10} - \frac{1}{64} a^{9} - \frac{173678957}{4209747840} a^{8} + \frac{7}{64} a^{7} + \frac{11401783}{467749760} a^{6} + \frac{5}{24} a^{5} + \frac{52031449}{350812320} a^{4} - \frac{19}{48} a^{3} + \frac{11633479}{116937440} a^{2} + \frac{5}{16} a + \frac{384234889}{1052436960}$, $\frac{1}{4209747840} a^{17} + \frac{28637}{467749760} a^{15} + \frac{407261}{350812320} a^{13} - \frac{1}{192} a^{12} + \frac{881853}{58468720} a^{11} - \frac{5}{192} a^{10} - \frac{173678957}{4209747840} a^{9} - \frac{1}{64} a^{8} + \frac{11401783}{467749760} a^{7} - \frac{9}{64} a^{6} + \frac{52031449}{350812320} a^{5} - \frac{7}{24} a^{4} - \frac{46835241}{116937440} a^{3} - \frac{7}{48} a^{2} + \frac{384234889}{1052436960} a + \frac{5}{16}$

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

Class group and class number

$C_{3}\times C_{3}$, which has order $9$

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:  \( 161496.647601 \)
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

\(\Q(\sqrt{-307}) \), 3.1.1228.3 x3, 3.1.307.1 x3, 3.1.1228.1 x3, 3.1.1228.2 x3, 6.0.462951088.2, 6.0.28934443.1, 6.0.462951088.1, 6.0.462951088.3, 9.1.568503936064.1 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 ${\href{/LocalNumberField/3.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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}$
307Data not computed