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Properties

Label 18.0.25815178338...6864.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{44}$
Root discriminant $29.33$
Ramified primes $2, 3$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3\times D_9$ (as 18T19)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![12301, -28566, 32643, -33822, 34641, -28674, 21528, -15390, 10044, -5760, 3123, -1620, 699, -270, 135, -18, 18, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 18*x^16 - 18*x^15 + 135*x^14 - 270*x^13 + 699*x^12 - 1620*x^11 + 3123*x^10 - 5760*x^9 + 10044*x^8 - 15390*x^7 + 21528*x^6 - 28674*x^5 + 34641*x^4 - 33822*x^3 + 32643*x^2 - 28566*x + 12301)
 
gp: K = bnfinit(x^18 + 18*x^16 - 18*x^15 + 135*x^14 - 270*x^13 + 699*x^12 - 1620*x^11 + 3123*x^10 - 5760*x^9 + 10044*x^8 - 15390*x^7 + 21528*x^6 - 28674*x^5 + 34641*x^4 - 33822*x^3 + 32643*x^2 - 28566*x + 12301, 1)
 

Normalized defining polynomial

\( x^{18} + 18 x^{16} - 18 x^{15} + 135 x^{14} - 270 x^{13} + 699 x^{12} - 1620 x^{11} + 3123 x^{10} - 5760 x^{9} + 10044 x^{8} - 15390 x^{7} + 21528 x^{6} - 28674 x^{5} + 34641 x^{4} - 33822 x^{3} + 32643 x^{2} - 28566 x + 12301 \)

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:  \(-258151783382020583032356864=-\,2^{18}\cdot 3^{44}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.33$
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:  $2, 3$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{4} a$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{16} a^{12} - \frac{1}{8} a^{9} + \frac{1}{8} a^{8} + \frac{1}{8} a^{7} - \frac{1}{8} a^{6} + \frac{1}{8} a^{5} - \frac{3}{16} a^{4} + \frac{1}{4} a^{3} - \frac{3}{8} a^{2} + \frac{3}{8} a - \frac{3}{16}$, $\frac{1}{16} a^{13} - \frac{1}{8} a^{10} - \frac{1}{8} a^{9} + \frac{1}{8} a^{8} + \frac{1}{8} a^{7} - \frac{1}{8} a^{6} + \frac{1}{16} a^{5} - \frac{1}{4} a^{4} + \frac{1}{8} a^{3} + \frac{1}{8} a^{2} - \frac{7}{16} a - \frac{1}{4}$, $\frac{1}{16} a^{14} - \frac{1}{8} a^{11} - \frac{1}{8} a^{10} - \frac{1}{8} a^{9} + \frac{1}{8} a^{8} + \frac{1}{8} a^{7} - \frac{3}{16} a^{6} - \frac{3}{8} a^{4} - \frac{3}{8} a^{3} + \frac{5}{16} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{112784} a^{15} + \frac{15}{112784} a^{13} + \frac{48}{7049} a^{12} + \frac{45}{56392} a^{11} + \frac{576}{7049} a^{10} + \frac{307}{7049} a^{9} - \frac{1865}{14098} a^{8} - \frac{14209}{112784} a^{7} + \frac{11441}{56392} a^{6} + \frac{12793}{112784} a^{5} - \frac{4051}{28196} a^{4} - \frac{34745}{112784} a^{3} - \frac{16515}{56392} a^{2} - \frac{29101}{112784} a - \frac{3483}{56392}$, $\frac{1}{112784} a^{16} + \frac{15}{112784} a^{14} + \frac{48}{7049} a^{13} + \frac{45}{56392} a^{12} + \frac{576}{7049} a^{11} + \frac{307}{7049} a^{10} + \frac{3319}{28196} a^{9} - \frac{14209}{112784} a^{8} - \frac{2657}{56392} a^{7} - \frac{15403}{112784} a^{6} - \frac{2775}{7049} a^{5} + \frac{21647}{112784} a^{4} + \frac{11681}{56392} a^{3} + \frac{55487}{112784} a^{2} + \frac{10615}{56392} a - \frac{1}{4}$, $\frac{1}{467038544} a^{17} - \frac{13}{66719792} a^{16} + \frac{17}{467038544} a^{15} - \frac{11405879}{467038544} a^{14} - \frac{342785}{12290488} a^{13} - \frac{1960263}{66719792} a^{12} + \frac{5106555}{116759636} a^{11} - \frac{5313989}{58379818} a^{10} + \frac{14732337}{467038544} a^{9} - \frac{11604219}{66719792} a^{8} + \frac{12155191}{467038544} a^{7} - \frac{88147459}{467038544} a^{6} - \frac{195032519}{467038544} a^{5} - \frac{189097}{5695592} a^{4} + \frac{204339363}{467038544} a^{3} + \frac{1524559}{24580976} a^{2} + \frac{16665309}{58379818} a - \frac{89952123}{467038544}$

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:  $8$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{81}{56392} a^{15} + \frac{1215}{56392} a^{13} - \frac{1233}{56392} a^{12} + \frac{3645}{28196} a^{11} - \frac{3699}{14098} a^{10} + \frac{7831}{14098} a^{9} - \frac{33291}{28196} a^{8} + \frac{117891}{56392} a^{7} - \frac{88335}{28196} a^{6} + \frac{274941}{56392} a^{5} - \frac{360909}{56392} a^{4} + \frac{399999}{56392} a^{3} - \frac{195777}{28196} a^{2} + \frac{434223}{56392} a - \frac{247041}{56392} \) (order $4$)
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:  \( 1082911.90412 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times D_9$ (as 18T19):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 18 conjugacy class representatives for $C_3\times D_9$
Character table for $C_3\times D_9$

Intermediate fields

\(\Q(\sqrt{-1}) \), 3.1.324.1 x3, 6.0.419904.2

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

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 ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{9}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$2$2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
3Data not computed