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Properties

Label 18.18.1493859726...9808.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{18}\cdot 3^{27}\cdot 73^{3}\cdot 577^{3}$
Root discriminant $61.30$
Ramified primes $2, 3, 73, 577$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 18T285

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1137267, 0, 3914973, 0, -4873932, 0, 3014559, 0, -1039797, 0, 210438, 0, -25341, 0, 1773, 0, -66, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 66*x^16 + 1773*x^14 - 25341*x^12 + 210438*x^10 - 1039797*x^8 + 3014559*x^6 - 4873932*x^4 + 3914973*x^2 - 1137267)
 
gp: K = bnfinit(x^18 - 66*x^16 + 1773*x^14 - 25341*x^12 + 210438*x^10 - 1039797*x^8 + 3014559*x^6 - 4873932*x^4 + 3914973*x^2 - 1137267, 1)
 

Normalized defining polynomial

\( x^{18} - 66 x^{16} + 1773 x^{14} - 25341 x^{12} + 210438 x^{10} - 1039797 x^{8} + 3014559 x^{6} - 4873932 x^{4} + 3914973 x^{2} - 1137267 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(149385972680924763865244423159808=2^{18}\cdot 3^{27}\cdot 73^{3}\cdot 577^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $61.30$
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:  $2, 3, 73, 577$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6}$, $\frac{1}{3} a^{7}$, $\frac{1}{3} a^{8}$, $\frac{1}{3} a^{9}$, $\frac{1}{3} a^{10}$, $\frac{1}{3} a^{11}$, $\frac{1}{9} a^{12}$, $\frac{1}{9} a^{13}$, $\frac{1}{171} a^{14} - \frac{1}{57} a^{12} + \frac{7}{57} a^{10} - \frac{2}{57} a^{8} + \frac{8}{57} a^{6} - \frac{7}{19} a^{2} + \frac{5}{19}$, $\frac{1}{171} a^{15} - \frac{1}{57} a^{13} + \frac{7}{57} a^{11} - \frac{2}{57} a^{9} + \frac{8}{57} a^{7} - \frac{7}{19} a^{3} + \frac{5}{19} a$, $\frac{1}{114773713002981} a^{16} - \frac{58551538859}{38257904334327} a^{14} - \frac{2260051204901}{114773713002981} a^{12} + \frac{974399566003}{12752634778109} a^{10} + \frac{3319563494143}{38257904334327} a^{8} - \frac{2280660605746}{38257904334327} a^{6} - \frac{2352831666198}{12752634778109} a^{4} + \frac{4871181438057}{12752634778109} a^{2} - \frac{1405605704788}{12752634778109}$, $\frac{1}{114773713002981} a^{17} - \frac{58551538859}{38257904334327} a^{15} - \frac{2260051204901}{114773713002981} a^{13} + \frac{974399566003}{12752634778109} a^{11} + \frac{3319563494143}{38257904334327} a^{9} - \frac{2280660605746}{38257904334327} a^{7} - \frac{2352831666198}{12752634778109} a^{5} + \frac{4871181438057}{12752634778109} a^{3} - \frac{1405605704788}{12752634778109} a$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $17$
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:  \( 9539915079.4 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T285:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1296
The 34 conjugacy class representatives for t18n285
Character table for t18n285 is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\), 6.6.53060329152.1, 9.9.22384826361.1

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

Sibling fields

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 $18$ $18$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ $18$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
2Data not computed
3Data not computed
73Data not computed
577Data not computed