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Properties

Label 18.6.12178167588...9257.2
Degree $18$
Signature $[6, 6]$
Discriminant $3^{24}\cdot 73^{3}\cdot 577^{4}$
Root discriminant $36.33$
Ramified primes $3, 73, 577$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T765

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![53, -4251, -153, 8128, 2427, 2775, -3302, -120, 126, -1128, 702, -120, 19, 3, 6, -23, 9, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 9*x^16 - 23*x^15 + 6*x^14 + 3*x^13 + 19*x^12 - 120*x^11 + 702*x^10 - 1128*x^9 + 126*x^8 - 120*x^7 - 3302*x^6 + 2775*x^5 + 2427*x^4 + 8128*x^3 - 153*x^2 - 4251*x + 53)
 
gp: K = bnfinit(x^18 - 3*x^17 + 9*x^16 - 23*x^15 + 6*x^14 + 3*x^13 + 19*x^12 - 120*x^11 + 702*x^10 - 1128*x^9 + 126*x^8 - 120*x^7 - 3302*x^6 + 2775*x^5 + 2427*x^4 + 8128*x^3 - 153*x^2 - 4251*x + 53, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 9 x^{16} - 23 x^{15} + 6 x^{14} + 3 x^{13} + 19 x^{12} - 120 x^{11} + 702 x^{10} - 1128 x^{9} + 126 x^{8} - 120 x^{7} - 3302 x^{6} + 2775 x^{5} + 2427 x^{4} + 8128 x^{3} - 153 x^{2} - 4251 x + 53 \)

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:  \(12178167588536804870357659257=3^{24}\cdot 73^{3}\cdot 577^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.33$
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:  $3, 73, 577$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{9} - \frac{1}{3} a^{6} - \frac{1}{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{10} - \frac{1}{3} a^{7} - \frac{1}{3} a$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{11} - \frac{1}{3} a^{8} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{15} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{9} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{2}{9} a^{6} + \frac{1}{3} a^{5} + \frac{2}{9} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{2}{9}$, $\frac{1}{9} a^{16} + \frac{1}{3} a^{11} + \frac{1}{9} a^{10} - \frac{1}{3} a^{8} + \frac{2}{9} a^{7} + \frac{2}{9} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{2}{9} a - \frac{1}{3}$, $\frac{1}{3774229483728472231525296776719209} a^{17} - \frac{69098343444462851834123878105585}{3774229483728472231525296776719209} a^{16} - \frac{7813700672355341468454262855957}{419358831525385803502810752968801} a^{15} - \frac{55124816868139805740699637487547}{1258076494576157410508432258906403} a^{14} - \frac{12498459221741546958316665073755}{419358831525385803502810752968801} a^{13} - \frac{28385311647846994605845486492100}{419358831525385803502810752968801} a^{12} + \frac{5073781090167006271063225433263}{3774229483728472231525296776719209} a^{11} + \frac{732347445379479880972483157032940}{3774229483728472231525296776719209} a^{10} - \frac{191311760318106036948237162878880}{419358831525385803502810752968801} a^{9} - \frac{1132522982759008760371840792590715}{3774229483728472231525296776719209} a^{8} - \frac{64796643779231184677559412895555}{3774229483728472231525296776719209} a^{7} - \frac{61534850026435467433792508869973}{1258076494576157410508432258906403} a^{6} - \frac{1656784179100501143355439578783627}{3774229483728472231525296776719209} a^{5} + \frac{1762997430385388573965496403895165}{3774229483728472231525296776719209} a^{4} - \frac{122489778962271803264240682180203}{419358831525385803502810752968801} a^{3} + \frac{361394630502240459932234437884074}{3774229483728472231525296776719209} a^{2} - \frac{605835081131006722078328419580753}{3774229483728472231525296776719209} a + \frac{174579243599187267231234723905255}{1258076494576157410508432258906403}$

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:  \( 9553128.96088 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T765:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 82944
The 110 conjugacy class representatives for t18n765 are not computed
Character table for t18n765 is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\), 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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ R $18$ $18$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ $18$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ $18$ $18$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$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$73.2.1.2$x^{2} + 365$$2$$1$$1$$C_2$$[\ ]_{2}$
73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.4.2.1$x^{4} + 1533 x^{2} + 644809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
73.4.0.1$x^{4} - x + 13$$1$$4$$0$$C_4$$[\ ]^{4}$
73.6.0.1$x^{6} - x + 5$$1$$6$$0$$C_6$$[\ ]^{6}$
577Data not computed