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Properties

Label 18.2.14701569336...8096.1
Degree $18$
Signature $[2, 8]$
Discriminant $2^{16}\cdot 3^{33}\cdot 7^{9}$
Root discriminant $36.72$
Ramified primes $2, 3, 7$
Class number $1$ (GRH)
Class group Trivial (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![-3485, 25869, -96759, 236262, -408012, 512586, -476778, 339330, -197937, 102799, -47595, 17640, -4680, 852, -72, -48, 33, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 33*x^16 - 48*x^15 - 72*x^14 + 852*x^13 - 4680*x^12 + 17640*x^11 - 47595*x^10 + 102799*x^9 - 197937*x^8 + 339330*x^7 - 476778*x^6 + 512586*x^5 - 408012*x^4 + 236262*x^3 - 96759*x^2 + 25869*x - 3485)
 
gp: K = bnfinit(x^18 - 9*x^17 + 33*x^16 - 48*x^15 - 72*x^14 + 852*x^13 - 4680*x^12 + 17640*x^11 - 47595*x^10 + 102799*x^9 - 197937*x^8 + 339330*x^7 - 476778*x^6 + 512586*x^5 - 408012*x^4 + 236262*x^3 - 96759*x^2 + 25869*x - 3485, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 33 x^{16} - 48 x^{15} - 72 x^{14} + 852 x^{13} - 4680 x^{12} + 17640 x^{11} - 47595 x^{10} + 102799 x^{9} - 197937 x^{8} + 339330 x^{7} - 476778 x^{6} + 512586 x^{5} - 408012 x^{4} + 236262 x^{3} - 96759 x^{2} + 25869 x - 3485 \)

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:  \(14701569336408730423883268096=2^{16}\cdot 3^{33}\cdot 7^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.72$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{71} a^{16} - \frac{26}{71} a^{15} - \frac{2}{71} a^{14} + \frac{34}{71} a^{13} + \frac{20}{71} a^{12} - \frac{15}{71} a^{11} + \frac{22}{71} a^{10} - \frac{3}{71} a^{9} - \frac{31}{71} a^{8} + \frac{32}{71} a^{7} - \frac{17}{71} a^{6} + \frac{27}{71} a^{5} - \frac{31}{71} a^{4} - \frac{32}{71} a^{3} + \frac{20}{71} a^{2} - \frac{12}{71} a - \frac{21}{71}$, $\frac{1}{8666949070927828160319456359} a^{17} + \frac{54073341350333403569006481}{8666949070927828160319456359} a^{16} + \frac{1922247449493940372289439597}{8666949070927828160319456359} a^{15} + \frac{3823538456458683853283466398}{8666949070927828160319456359} a^{14} + \frac{3626266861461084726184350474}{8666949070927828160319456359} a^{13} - \frac{2126177559544473201530108770}{8666949070927828160319456359} a^{12} - \frac{1730145399525563288990709379}{8666949070927828160319456359} a^{11} + \frac{406974069927865544904857018}{8666949070927828160319456359} a^{10} + \frac{4312204287840408864343080842}{8666949070927828160319456359} a^{9} - \frac{3791444949763164486412623621}{8666949070927828160319456359} a^{8} - \frac{1918675176585748209323811706}{8666949070927828160319456359} a^{7} - \frac{608420191535782317024070055}{8666949070927828160319456359} a^{6} + \frac{2124571894468484116032890566}{8666949070927828160319456359} a^{5} + \frac{3061749097833089698698578847}{8666949070927828160319456359} a^{4} - \frac{1823857964087745357647853969}{8666949070927828160319456359} a^{3} - \frac{773038350701421489142090619}{8666949070927828160319456359} a^{2} + \frac{3646768949566334036852327028}{8666949070927828160319456359} a - \frac{377552171532217080323525417}{8666949070927828160319456359}$

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:  $9$
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:  \( 14488152.048552748 \) (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

\(\Q(\sqrt{21}) \), 3.1.108.1, 6.2.12002256.1, 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ $18$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ $18$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
2Data not computed
3Data not computed
7Data not computed