↑

Properties

Label 18.0.20001322776...2016.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{24}\cdot 3^{45}\cdot 7^{9}$
Root discriminant $103.93$
Ramified primes $2, 3, 7$
Class number $36$ (GRH)
Class group $[2, 18]$ (GRH)
Galois group $C_2\times He_3:C_2$ (as 18T41)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![163085133, -230812092, 248784480, -168009228, 93863448, -36738198, 16914231, -3068604, 1659042, 187920, 112014, 35154, 7191, 0, 126, -108, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 108*x^15 + 126*x^14 + 7191*x^12 + 35154*x^11 + 112014*x^10 + 187920*x^9 + 1659042*x^8 - 3068604*x^7 + 16914231*x^6 - 36738198*x^5 + 93863448*x^4 - 168009228*x^3 + 248784480*x^2 - 230812092*x + 163085133)
 
gp: K = bnfinit(x^18 - 108*x^15 + 126*x^14 + 7191*x^12 + 35154*x^11 + 112014*x^10 + 187920*x^9 + 1659042*x^8 - 3068604*x^7 + 16914231*x^6 - 36738198*x^5 + 93863448*x^4 - 168009228*x^3 + 248784480*x^2 - 230812092*x + 163085133, 1)
 

Normalized defining polynomial

\( x^{18} - 108 x^{15} + 126 x^{14} + 7191 x^{12} + 35154 x^{11} + 112014 x^{10} + 187920 x^{9} + 1659042 x^{8} - 3068604 x^{7} + 16914231 x^{6} - 36738198 x^{5} + 93863448 x^{4} - 168009228 x^{3} + 248784480 x^{2} - 230812092 x + 163085133 \)

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:  \(-2000132277685860378930930657332822016=-\,2^{24}\cdot 3^{45}\cdot 7^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $103.93$
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:  $2, 3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} + \frac{1}{5} a^{9} + \frac{1}{5} a^{8} + \frac{1}{5} a^{6} - \frac{2}{5} a^{5} - \frac{1}{5} a^{4} + \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{5} a^{13} + \frac{1}{5} a^{10} + \frac{1}{5} a^{9} + \frac{1}{5} a^{7} - \frac{2}{5} a^{6} - \frac{1}{5} a^{5} + \frac{2}{5} a^{2} + \frac{2}{5} a$, $\frac{1}{5} a^{14} + \frac{1}{5} a^{11} + \frac{1}{5} a^{10} + \frac{1}{5} a^{8} - \frac{2}{5} a^{7} - \frac{1}{5} a^{6} + \frac{2}{5} a^{3} + \frac{2}{5} a^{2}$, $\frac{1}{25} a^{15} - \frac{1}{25} a^{12} + \frac{1}{25} a^{11} + \frac{9}{25} a^{9} + \frac{11}{25} a^{8} - \frac{11}{25} a^{7} + \frac{8}{25} a^{6} - \frac{6}{25} a^{5} - \frac{6}{25} a^{4} + \frac{12}{25} a^{3} + \frac{2}{5} a^{2} + \frac{6}{25} a + \frac{6}{25}$, $\frac{1}{25} a^{16} - \frac{1}{25} a^{13} + \frac{1}{25} a^{12} + \frac{9}{25} a^{10} + \frac{11}{25} a^{9} - \frac{11}{25} a^{8} + \frac{8}{25} a^{7} - \frac{6}{25} a^{6} - \frac{6}{25} a^{5} + \frac{12}{25} a^{4} + \frac{2}{5} a^{3} + \frac{6}{25} a^{2} + \frac{6}{25} a$, $\frac{1}{222364464966156604517205216279754932868086124980439315691179516080575} a^{17} - \frac{167907812868832211338451944430214247795082894384510136874522362059}{8894578598646264180688208651190197314723444999217572627647180643223} a^{16} - \frac{4304771605276563476203262759985758523351972132335388610392725207356}{222364464966156604517205216279754932868086124980439315691179516080575} a^{15} - \frac{15599339817807545324276793153520345771318942289731744030987848978266}{222364464966156604517205216279754932868086124980439315691179516080575} a^{14} + \frac{11363059128700247880071938344738326476240483640159126040376863682301}{222364464966156604517205216279754932868086124980439315691179516080575} a^{13} - \frac{13057861786516545924752888447129180343378630326756718768354889365174}{222364464966156604517205216279754932868086124980439315691179516080575} a^{12} + \frac{86722574816431163629802533361235242752086260868273498261951587499}{282546969461444224291239156645177805423235228691790744207343730725} a^{11} + \frac{3417480497004867660145534900010899754356354413738778517423929296421}{222364464966156604517205216279754932868086124980439315691179516080575} a^{10} - \frac{14336360937048084229507525536416798848746875654939351280420473949814}{44472892993231320903441043255950986573617224996087863138235903216115} a^{9} + \frac{97789032232579044167207413990522009027705167570143905129283700285697}{222364464966156604517205216279754932868086124980439315691179516080575} a^{8} + \frac{6954626119227671648422896998915244847240278623270731013127823836193}{44472892993231320903441043255950986573617224996087863138235903216115} a^{7} + \frac{48521085954146159824552753236248623652930546246342409357295538928431}{222364464966156604517205216279754932868086124980439315691179516080575} a^{6} + \frac{96796737839799993946464232338220707063789124735590017973658277909583}{222364464966156604517205216279754932868086124980439315691179516080575} a^{5} + \frac{50895325879121452837166982821455447478515871637536614734576516087976}{222364464966156604517205216279754932868086124980439315691179516080575} a^{4} - \frac{10272316598710633382991469535972754495135827948033777794824544320421}{222364464966156604517205216279754932868086124980439315691179516080575} a^{3} - \frac{83802908084161118634278806592739635045363657148886265975382215750409}{222364464966156604517205216279754932868086124980439315691179516080575} a^{2} - \frac{40848450098327837075456047186518258784891487023865856222566866410346}{222364464966156604517205216279754932868086124980439315691179516080575} a - \frac{41218285592765433184595443996648881460038899122709475557620260475096}{222364464966156604517205216279754932868086124980439315691179516080575}$

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

Class group and class number

$C_{2}\times C_{18}$, which has order $36$ (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:  \( 7090895688.796117 \) (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

\(\Q(\sqrt{-21}) \), 3.1.243.1, 6.0.3888730944.9, 9.1.2008387814976.4

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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
$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
$7$7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$