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Properties

Label 18.2.12100864846...0000.1
Degree $18$
Signature $[2, 8]$
Discriminant $2^{24}\cdot 3^{45}\cdot 5^{12}$
Root discriminant $114.86$
Ramified primes $2, 3, 5$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_2\times He_3:C_2$ (as 18T41)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-474743, -937800, -2316780, -4907100, -6882300, -7134930, -5782233, -3258900, -1471230, -402550, -85410, -22050, -651, 0, 90, -30, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 30*x^15 + 90*x^14 - 651*x^12 - 22050*x^11 - 85410*x^10 - 402550*x^9 - 1471230*x^8 - 3258900*x^7 - 5782233*x^6 - 7134930*x^5 - 6882300*x^4 - 4907100*x^3 - 2316780*x^2 - 937800*x - 474743)
 
gp: K = bnfinit(x^18 - 30*x^15 + 90*x^14 - 651*x^12 - 22050*x^11 - 85410*x^10 - 402550*x^9 - 1471230*x^8 - 3258900*x^7 - 5782233*x^6 - 7134930*x^5 - 6882300*x^4 - 4907100*x^3 - 2316780*x^2 - 937800*x - 474743, 1)
 

Normalized defining polynomial

\( x^{18} - 30 x^{15} + 90 x^{14} - 651 x^{12} - 22050 x^{11} - 85410 x^{10} - 402550 x^{9} - 1471230 x^{8} - 3258900 x^{7} - 5782233 x^{6} - 7134930 x^{5} - 6882300 x^{4} - 4907100 x^{3} - 2316780 x^{2} - 937800 x - 474743 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(12100864846032214829641728000000000000=2^{24}\cdot 3^{45}\cdot 5^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $114.86$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{14} - \frac{1}{2} a^{13} - \frac{1}{4} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{6349968049919550487368880131349060285095249904734554804271208} a^{17} - \frac{748584293198491772874930524684057988406454775217739825694113}{6349968049919550487368880131349060285095249904734554804271208} a^{16} + \frac{1232537965361293184974453759125220148947463124453547414956671}{6349968049919550487368880131349060285095249904734554804271208} a^{15} + \frac{511038486497265284095565186231427983793721217428665364236795}{6349968049919550487368880131349060285095249904734554804271208} a^{14} - \frac{2863517898453000776421097178534089680989777132928154968074383}{6349968049919550487368880131349060285095249904734554804271208} a^{13} + \frac{475524997643202240507654295879478777326078264835483107761851}{6349968049919550487368880131349060285095249904734554804271208} a^{12} - \frac{697498654310107753202436614476423465672111910462317035047247}{1587492012479887621842220032837265071273812476183638701067802} a^{11} + \frac{25209816220062940724614570163653464742874118501398301459579}{3174984024959775243684440065674530142547624952367277402135604} a^{10} + \frac{687521266264886398636669802761925493760661534425755178942853}{1587492012479887621842220032837265071273812476183638701067802} a^{9} - \frac{1133321381386171170878848889076984676761058519938050267806589}{3174984024959775243684440065674530142547624952367277402135604} a^{8} - \frac{533968431468484278565771953096150766084899561037286389767383}{1587492012479887621842220032837265071273812476183638701067802} a^{7} - \frac{792111434730516059513783930292027723617828574398582522303767}{1587492012479887621842220032837265071273812476183638701067802} a^{6} - \frac{2636248017082592273690152589158255888032915712078159417265921}{6349968049919550487368880131349060285095249904734554804271208} a^{5} - \frac{2351201147598145057846542844976030653089825946611683057223213}{6349968049919550487368880131349060285095249904734554804271208} a^{4} + \frac{2016006215185743359444384943105443509943735998316702451211791}{6349968049919550487368880131349060285095249904734554804271208} a^{3} + \frac{2193535079072852010854789115608162796318343788189329503621389}{6349968049919550487368880131349060285095249904734554804271208} a^{2} - \frac{1044359954676844708571539674842064052269636216570560564631383}{6349968049919550487368880131349060285095249904734554804271208} a + \frac{1742830289917366521941279944768022994232479793271748819882031}{6349968049919550487368880131349060285095249904734554804271208}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  $9$
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:  \( 253112842543.77023 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times He_3:C_2$ (as 18T41):

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 $C_2\times He_3:C_2$
Character table for $C_2\times He_3:C_2$

Intermediate fields

\(\Q(\sqrt{3}) \), 3.1.243.1, 6.2.11337408.2, 9.1.31381059609000000.8

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 R ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.6.8.3$x^{6} + 2 x^{3} + 6$$6$$1$$8$$D_{6}$$[2]_{3}^{2}$
2.12.16.3$x^{12} - 30 x^{10} - 5 x^{8} + 19 x^{4} + 30 x^{2} + 1$$6$$2$$16$$C_6\times S_3$$[2]_{3}^{6}$
3Data not computed
$5$5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
5.6.4.2$x^{6} - 5 x^{3} + 50$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
5.6.4.2$x^{6} - 5 x^{3} + 50$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$