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Properties

Label 18.0.38634162941...9344.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{16}\cdot 3^{33}\cdot 13^{9}$
Root discriminant $50.04$
Ramified primes $2, 3, 13$
Class number $12$ (GRH)
Class group $[12]$ (GRH)
Galois group $C_2\times D_9:C_3$ (as 18T45)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![252318, 217134, 645660, 160920, 641376, -220644, 228612, -187128, 189630, -114453, 50337, -12114, 1656, -432, 378, -162, 48, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 48*x^16 - 162*x^15 + 378*x^14 - 432*x^13 + 1656*x^12 - 12114*x^11 + 50337*x^10 - 114453*x^9 + 189630*x^8 - 187128*x^7 + 228612*x^6 - 220644*x^5 + 641376*x^4 + 160920*x^3 + 645660*x^2 + 217134*x + 252318)
 
gp: K = bnfinit(x^18 - 9*x^17 + 48*x^16 - 162*x^15 + 378*x^14 - 432*x^13 + 1656*x^12 - 12114*x^11 + 50337*x^10 - 114453*x^9 + 189630*x^8 - 187128*x^7 + 228612*x^6 - 220644*x^5 + 641376*x^4 + 160920*x^3 + 645660*x^2 + 217134*x + 252318, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 48 x^{16} - 162 x^{15} + 378 x^{14} - 432 x^{13} + 1656 x^{12} - 12114 x^{11} + 50337 x^{10} - 114453 x^{9} + 189630 x^{8} - 187128 x^{7} + 228612 x^{6} - 220644 x^{5} + 641376 x^{4} + 160920 x^{3} + 645660 x^{2} + 217134 x + 252318 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-3863416294113743236181988409344=-\,2^{16}\cdot 3^{33}\cdot 13^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $50.04$
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:  $2, 3, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{4} a^{12} + \frac{1}{4} a^{11} - \frac{1}{4} a^{10} + \frac{1}{4} a^{9} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{8} a^{13} - \frac{1}{4} a^{11} + \frac{1}{4} a^{10} + \frac{3}{8} a^{9} - \frac{1}{2} a^{8} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{8} a^{14} - \frac{1}{2} a^{11} + \frac{1}{8} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{5} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{32} a^{15} - \frac{1}{16} a^{14} - \frac{1}{32} a^{13} - \frac{1}{8} a^{12} + \frac{3}{32} a^{11} - \frac{7}{16} a^{10} - \frac{15}{32} a^{9} + \frac{3}{8} a^{8} - \frac{1}{2} a^{7} + \frac{3}{16} a^{6} + \frac{3}{8} a^{5} - \frac{3}{16} a^{4} - \frac{1}{2} a^{3} - \frac{7}{16} a^{2} - \frac{3}{8} a - \frac{1}{16}$, $\frac{1}{32} a^{16} - \frac{1}{32} a^{14} - \frac{1}{16} a^{13} + \frac{3}{32} a^{12} + \frac{1}{4} a^{11} - \frac{7}{32} a^{10} - \frac{3}{16} a^{9} - \frac{1}{4} a^{8} + \frac{3}{16} a^{7} - \frac{1}{4} a^{6} + \frac{5}{16} a^{5} - \frac{1}{8} a^{4} + \frac{1}{16} a^{3} - \frac{1}{4} a^{2} - \frac{1}{16} a + \frac{1}{8}$, $\frac{1}{1002885234048081538115430999046558526873319205587392} a^{17} - \frac{876547103248508644549071004215328264011843785035}{250721308512020384528857749761639631718329801396848} a^{16} - \frac{1142977613507586239297458011959959844150121963713}{501442617024040769057715499523279263436659602793696} a^{15} - \frac{1176777993094620303872081189297317565682998432749}{31340163564002548066107218720204953964791225174606} a^{14} - \frac{7345477734502223681087537088551325324387714417505}{125360654256010192264428874880819815859164900698424} a^{13} - \frac{5254261580495928946211430298203524969099361565221}{125360654256010192264428874880819815859164900698424} a^{12} + \frac{52078686512128970413804105951226856891372427585487}{501442617024040769057715499523279263436659602793696} a^{11} + \frac{4908253192935377446390278383615133779847050685619}{125360654256010192264428874880819815859164900698424} a^{10} - \frac{217696738505316134299584498689869386041625143627861}{1002885234048081538115430999046558526873319205587392} a^{9} + \frac{156328991548210336930393086920605618633805879647205}{501442617024040769057715499523279263436659602793696} a^{8} - \frac{8338738971906558899348294447983881359419809220939}{31340163564002548066107218720204953964791225174606} a^{7} + \frac{38223795926501713609004266806493567615190532182185}{250721308512020384528857749761639631718329801396848} a^{6} + \frac{24059227091605133271767392608506407448890202848507}{62680327128005096132214437440409907929582450349212} a^{5} - \frac{5239749916258526477680490291890523867696691738395}{15670081782001274033053609360102476982395612587303} a^{4} + \frac{22730584179151282580036768268572871965446115715449}{62680327128005096132214437440409907929582450349212} a^{3} - \frac{115259119538127617723256551527165699032574741565793}{250721308512020384528857749761639631718329801396848} a^{2} + \frac{21867943656303451167411319878184812476167962156809}{62680327128005096132214437440409907929582450349212} a + \frac{77726108843346094196788986406314242632692622793101}{501442617024040769057715499523279263436659602793696}$

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

Class group and class number

$C_{12}$, which has order $12$ (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:  $8$
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:  \( 44536004.20913794 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times D_9:C_3$ (as 18T45):

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 $C_2\times D_9:C_3$
Character table for $C_2\times D_9:C_3$

Intermediate fields

\(\Q(\sqrt{-39}) \), 3.1.108.1, 6.0.76877424.1, 9.1.11019960576.1

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

Sibling fields

Degree 18 sibling: 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ $18$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ $18$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ $18$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.9.8.1$x^{9} - 2$$9$$1$$8$$(C_9:C_3):C_2$$[\ ]_{9}^{6}$
2.9.8.1$x^{9} - 2$$9$$1$$8$$(C_9:C_3):C_2$$[\ ]_{9}^{6}$
3Data not computed
13Data not computed