↑

Properties

Label 18.0.37329525519...3104.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{24}\cdot 19^{9}$
Root discriminant $29.94$
Ramified primes $2, 3, 19$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_3^2 : C_2$ (as 18T4)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6156, 0, 17820, 0, 12240, 0, 3469, 0, -1026, 0, -813, 0, 36, 0, 99, 0, 18, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 18*x^16 + 99*x^14 + 36*x^12 - 813*x^10 - 1026*x^8 + 3469*x^6 + 12240*x^4 + 17820*x^2 + 6156)
 
gp: K = bnfinit(x^18 + 18*x^16 + 99*x^14 + 36*x^12 - 813*x^10 - 1026*x^8 + 3469*x^6 + 12240*x^4 + 17820*x^2 + 6156, 1)
 

Normalized defining polynomial

\( x^{18} + 18 x^{16} + 99 x^{14} + 36 x^{12} - 813 x^{10} - 1026 x^{8} + 3469 x^{6} + 12240 x^{4} + 17820 x^{2} + 6156 \)

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:  \(-373295255190912955087663104=-\,2^{12}\cdot 3^{24}\cdot 19^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.94$
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:  $2, 3, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4} a^{11} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{10} - \frac{1}{8} a^{8} + \frac{1}{8} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{11} - \frac{1}{8} a^{9} + \frac{1}{8} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{29640} a^{14} + \frac{21}{1976} a^{12} - \frac{33}{760} a^{10} - \frac{81}{9880} a^{8} - \frac{1}{4} a^{7} - \frac{737}{4940} a^{6} - \frac{1}{2} a^{5} - \frac{881}{2470} a^{4} - \frac{1}{4} a^{3} + \frac{613}{7410} a^{2} - \frac{1}{2} a + \frac{7}{130}$, $\frac{1}{177840} a^{15} - \frac{1}{59280} a^{14} - \frac{15}{247} a^{13} + \frac{113}{1976} a^{12} - \frac{53}{760} a^{11} + \frac{8}{95} a^{10} + \frac{151}{2470} a^{9} - \frac{577}{9880} a^{8} - \frac{12589}{59280} a^{7} - \frac{4701}{19760} a^{6} + \frac{59}{2470} a^{5} + \frac{529}{1235} a^{4} + \frac{613}{44460} a^{3} - \frac{613}{14820} a^{2} - \frac{41}{260} a + \frac{123}{260}$, $\frac{1}{125504355600} a^{16} - \frac{436327}{41834785200} a^{14} + \frac{125066801}{6972464200} a^{12} + \frac{2389207}{240429800} a^{10} - \frac{5566477}{71026800} a^{8} - \frac{10375447}{146788720} a^{6} - \frac{13606479179}{31376088900} a^{4} - \frac{1}{2} a^{3} + \frac{677099663}{2614674075} a^{2} - \frac{1}{2} a + \frac{38062091}{183485900}$, $\frac{1}{753026133600} a^{17} - \frac{1}{251008711200} a^{16} - \frac{436327}{251008711200} a^{15} - \frac{975103}{83669570400} a^{14} - \frac{1618049249}{41834785200} a^{13} + \frac{672391149}{13944928400} a^{12} + \frac{2389207}{1442578800} a^{11} - \frac{22003217}{480859600} a^{10} - \frac{41079877}{426160800} a^{9} + \frac{21962731}{207616800} a^{8} - \frac{10375447}{880732320} a^{7} - \frac{432646573}{5577971360} a^{6} - \frac{2881228477}{94128266700} a^{5} + \frac{163016206}{603386325} a^{4} + \frac{3968873401}{31376088900} a^{3} + \frac{413935727}{5229348150} a^{2} + \frac{135011297}{366971800} a - \frac{47942101}{366971800}$

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:  $8$
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:  \( 699865.028477 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3:S_3$ (as 18T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 6 conjugacy class representatives for $C_3^2 : C_2$
Character table for $C_3^2 : C_2$

Intermediate fields

\(\Q(\sqrt{-19}) \), 3.1.6156.2 x3, 3.1.6156.1 x3, 3.1.1539.1 x3, 3.1.76.1 x3, 6.0.720030384.1, 6.0.720030384.2, 6.0.45001899.2, 6.0.109744.2, 9.1.4432507043904.1 x9

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

Sibling fields

Degree 9 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 ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$3$3.6.8.5$x^{6} + 9 x^{2} + 9$$3$$2$$8$$S_3$$[2]^{2}$
3.6.8.5$x^{6} + 9 x^{2} + 9$$3$$2$$8$$S_3$$[2]^{2}$
3.6.8.5$x^{6} + 9 x^{2} + 9$$3$$2$$8$$S_3$$[2]^{2}$
$19$19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$