Properties

Label 18.6.12137764710...3169.1
Degree $18$
Signature $[6, 6]$
Discriminant $3^{24}\cdot 73^{10}$
Root discriminant $46.92$
Ramified primes $3, 73$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_6^2:C_3$ (as 18T48)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3691, -14862, -12165, 20953, 44793, 28032, -1448, -7242, -1536, -166, 1962, 210, 140, -204, 45, -50, 15, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 15*x^16 - 50*x^15 + 45*x^14 - 204*x^13 + 140*x^12 + 210*x^11 + 1962*x^10 - 166*x^9 - 1536*x^8 - 7242*x^7 - 1448*x^6 + 28032*x^5 + 44793*x^4 + 20953*x^3 - 12165*x^2 - 14862*x - 3691)
 
gp: K = bnfinit(x^18 - 3*x^17 + 15*x^16 - 50*x^15 + 45*x^14 - 204*x^13 + 140*x^12 + 210*x^11 + 1962*x^10 - 166*x^9 - 1536*x^8 - 7242*x^7 - 1448*x^6 + 28032*x^5 + 44793*x^4 + 20953*x^3 - 12165*x^2 - 14862*x - 3691, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 15 x^{16} - 50 x^{15} + 45 x^{14} - 204 x^{13} + 140 x^{12} + 210 x^{11} + 1962 x^{10} - 166 x^{9} - 1536 x^{8} - 7242 x^{7} - 1448 x^{6} + 28032 x^{5} + 44793 x^{4} + 20953 x^{3} - 12165 x^{2} - 14862 x - 3691 \)

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:  \(1213776471051948828443732093169=3^{24}\cdot 73^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $46.92$
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:  $3, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{3} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{6} + \frac{1}{4} a^{4} + \frac{1}{4} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{12} - \frac{1}{8} a^{11} - \frac{1}{8} a^{10} + \frac{1}{8} a^{9} + \frac{1}{8} a^{8} - \frac{1}{8} a^{7} - \frac{1}{8} a^{6} + \frac{1}{8} a^{5} - \frac{3}{8} a^{4} - \frac{3}{8} a^{3} + \frac{3}{8} a^{2} - \frac{1}{8} a - \frac{3}{8}$, $\frac{1}{8} a^{14} - \frac{1}{4} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{8}$, $\frac{1}{8} a^{15} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{1}{8} a - \frac{1}{4}$, $\frac{1}{8} a^{16} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} + \frac{1}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{719810801752868462662842910112704} a^{17} - \frac{7595138324982697780289069719773}{359905400876434231331421455056352} a^{16} - \frac{38644787744318261435073672247691}{719810801752868462662842910112704} a^{15} - \frac{13870482919229615540608207916429}{719810801752868462662842910112704} a^{14} - \frac{3453414297419609595066121369523}{89976350219108557832855363764088} a^{13} + \frac{7587005930004937858559417407863}{179952700438217115665710727528176} a^{12} - \frac{7497247652752576062194062335405}{89976350219108557832855363764088} a^{11} - \frac{21626317898293327831881643361547}{359905400876434231331421455056352} a^{10} - \frac{16345068226269875808495031914523}{179952700438217115665710727528176} a^{9} + \frac{613383782148307687681391796687}{359905400876434231331421455056352} a^{8} + \frac{40857463634218205947230275802639}{359905400876434231331421455056352} a^{7} + \frac{69028866880212001592376698006829}{179952700438217115665710727528176} a^{6} - \frac{56912897122915546085667330547765}{179952700438217115665710727528176} a^{5} - \frac{46476821982953717369235080077049}{179952700438217115665710727528176} a^{4} + \frac{252781306447230905503964380816021}{719810801752868462662842910112704} a^{3} - \frac{17215949775977662188065941615517}{359905400876434231331421455056352} a^{2} + \frac{76808309623903307330687067885809}{719810801752868462662842910112704} a + \frac{272491919549478809484224999605555}{719810801752868462662842910112704}$

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:  \( 208631462.937 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6^2:C_3$ (as 18T48):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 20 conjugacy class representatives for $C_6^2:C_3$
Character table for $C_6^2:C_3$

Intermediate fields

\(\Q(\zeta_{9})^+\), 6.2.34963569.1, 9.9.15091989595281.1

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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$3$3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
$73$73.6.0.1$x^{6} - x + 5$$1$$6$$0$$C_6$$[\ ]^{6}$
73.6.5.4$x^{6} + 365$$6$$1$$5$$C_6$$[\ ]_{6}$
73.6.5.2$x^{6} - 1825$$6$$1$$5$$C_6$$[\ ]_{6}$