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Properties

Label 18.6.43584805012...0000.1
Degree $18$
Signature $[6, 6]$
Discriminant $2^{12}\cdot 3^{20}\cdot 5^{15}$
Root discriminant $20.57$
Ramified primes $2, 3, 5$
Class number $1$ (GRH)
Class group Trivial (GRH)
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![1, 0, -42, 29, 165, -90, -285, 105, 195, -60, 48, 30, -111, 12, 45, -11, 0, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 11*x^15 + 45*x^14 + 12*x^13 - 111*x^12 + 30*x^11 + 48*x^10 - 60*x^9 + 195*x^8 + 105*x^7 - 285*x^6 - 90*x^5 + 165*x^4 + 29*x^3 - 42*x^2 + 1)
 
gp: K = bnfinit(x^18 - 3*x^17 - 11*x^15 + 45*x^14 + 12*x^13 - 111*x^12 + 30*x^11 + 48*x^10 - 60*x^9 + 195*x^8 + 105*x^7 - 285*x^6 - 90*x^5 + 165*x^4 + 29*x^3 - 42*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 11 x^{15} + 45 x^{14} + 12 x^{13} - 111 x^{12} + 30 x^{11} + 48 x^{10} - 60 x^{9} + 195 x^{8} + 105 x^{7} - 285 x^{6} - 90 x^{5} + 165 x^{4} + 29 x^{3} - 42 x^{2} + 1 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(435848050125000000000000=2^{12}\cdot 3^{20}\cdot 5^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.57$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} - \frac{1}{5} a^{11} - \frac{2}{5} a^{10} - \frac{1}{5} a^{7} + \frac{1}{5} a^{6} + \frac{2}{5} a^{5} - \frac{2}{5} a^{2} + \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{13} + \frac{2}{5} a^{11} - \frac{2}{5} a^{10} - \frac{1}{5} a^{8} - \frac{2}{5} a^{6} + \frac{2}{5} a^{5} - \frac{2}{5} a^{3} + \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{14} - \frac{1}{5} a^{10} - \frac{1}{5} a^{9} + \frac{1}{5} a^{5} - \frac{2}{5} a^{4} + \frac{2}{5}$, $\frac{1}{5} a^{15} - \frac{1}{5} a^{11} - \frac{1}{5} a^{10} + \frac{1}{5} a^{6} - \frac{2}{5} a^{5} + \frac{2}{5} a$, $\frac{1}{35} a^{16} - \frac{1}{35} a^{14} - \frac{3}{35} a^{12} - \frac{4}{35} a^{11} - \frac{3}{7} a^{10} - \frac{4}{35} a^{9} - \frac{1}{7} a^{8} + \frac{13}{35} a^{7} + \frac{1}{35} a^{6} - \frac{2}{7} a^{5} - \frac{8}{35} a^{4} + \frac{3}{7} a^{3} + \frac{1}{35} a^{2} - \frac{4}{35} a + \frac{3}{7}$, $\frac{1}{15590740940075} a^{17} + \frac{122159446329}{15590740940075} a^{16} - \frac{730066203812}{15590740940075} a^{15} + \frac{24666675231}{3118148188015} a^{14} - \frac{210312332491}{3118148188015} a^{13} + \frac{351261575187}{15590740940075} a^{12} + \frac{2894343158543}{15590740940075} a^{11} - \frac{4360413201264}{15590740940075} a^{10} + \frac{73497323828}{445449741145} a^{9} - \frac{112385146669}{445449741145} a^{8} + \frac{141505600101}{3118148188015} a^{7} - \frac{464108206601}{3118148188015} a^{6} + \frac{938540699519}{3118148188015} a^{5} - \frac{1190444889076}{3118148188015} a^{4} + \frac{83564506336}{623629637603} a^{3} - \frac{409396362238}{2227248705725} a^{2} + \frac{3697317323181}{15590740940075} a + \frac{5996826699112}{15590740940075}$

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:  $11$
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:  \( 90961.5357762 \) (assuming GRH)
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

\(\Q(\sqrt{5}) \), 3.1.135.1, 3.3.2700.1, 6.2.91125.1, 6.6.36450000.1, 9.3.295245000000.1

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 ${\href{/LocalNumberField/7.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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$2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
2.12.12.26$x^{12} - 162 x^{10} + 26423 x^{8} + 125508 x^{6} - 64481 x^{4} - 122498 x^{2} - 86071$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
$3$3.6.6.4$x^{6} + 3 x^{4} + 6 x^{3} + 9 x^{2} + 63 x + 9$$3$$2$$6$$D_{6}$$[3/2]_{2}^{2}$
3.12.14.11$x^{12} + 6 x^{11} + 21 x^{10} + 36 x^{9} + 30 x^{8} + 36 x^{7} + 3 x^{6} + 36 x^{5} + 27 x^{4} - 9 x^{2} + 36$$6$$2$$14$$D_6$$[3/2]_{2}^{2}$
$5$5.6.5.1$x^{6} - 5$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
5.12.10.1$x^{12} + 6 x^{11} + 27 x^{10} + 80 x^{9} + 195 x^{8} + 366 x^{7} + 571 x^{6} + 702 x^{5} + 1005 x^{4} + 1140 x^{3} + 357 x^{2} - 138 x + 44$$6$$2$$10$$D_6$$[\ ]_{6}^{2}$