↑

Properties

Label 18.0.11990701314...5739.4
Degree $18$
Signature $[0, 9]$
Discriminant $-\,7^{12}\cdot 59^{9}$
Root discriminant $28.11$
Ramified primes $7, 59$
Class number $9$
Class group $[9]$
Galois group $S_3 \times C_3$ (as 18T3)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![19629, 3645, 4272, 16859, 12837, -1335, 16855, -7645, 9424, -4561, 3227, -1532, 828, -329, 158, -39, 19, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 + 19*x^16 - 39*x^15 + 158*x^14 - 329*x^13 + 828*x^12 - 1532*x^11 + 3227*x^10 - 4561*x^9 + 9424*x^8 - 7645*x^7 + 16855*x^6 - 1335*x^5 + 12837*x^4 + 16859*x^3 + 4272*x^2 + 3645*x + 19629)
 
gp: K = bnfinit(x^18 - 2*x^17 + 19*x^16 - 39*x^15 + 158*x^14 - 329*x^13 + 828*x^12 - 1532*x^11 + 3227*x^10 - 4561*x^9 + 9424*x^8 - 7645*x^7 + 16855*x^6 - 1335*x^5 + 12837*x^4 + 16859*x^3 + 4272*x^2 + 3645*x + 19629, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{17} + 19 x^{16} - 39 x^{15} + 158 x^{14} - 329 x^{13} + 828 x^{12} - 1532 x^{11} + 3227 x^{10} - 4561 x^{9} + 9424 x^{8} - 7645 x^{7} + 16855 x^{6} - 1335 x^{5} + 12837 x^{4} + 16859 x^{3} + 4272 x^{2} + 3645 x + 19629 \)

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:  \(-119907013147065124216135739=-\,7^{12}\cdot 59^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.11$
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:  $7, 59$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{41} a^{14} + \frac{1}{41} a^{13} - \frac{13}{41} a^{12} - \frac{17}{41} a^{11} - \frac{9}{41} a^{10} + \frac{5}{41} a^{9} - \frac{3}{41} a^{8} - \frac{15}{41} a^{7} + \frac{12}{41} a^{6} + \frac{3}{41} a^{5} - \frac{20}{41} a^{4} - \frac{15}{41} a^{3} - \frac{5}{41} a^{2} + \frac{10}{41} a + \frac{14}{41}$, $\frac{1}{41} a^{15} - \frac{14}{41} a^{13} - \frac{4}{41} a^{12} + \frac{8}{41} a^{11} + \frac{14}{41} a^{10} - \frac{8}{41} a^{9} - \frac{12}{41} a^{8} - \frac{14}{41} a^{7} - \frac{9}{41} a^{6} + \frac{18}{41} a^{5} + \frac{5}{41} a^{4} + \frac{10}{41} a^{3} + \frac{15}{41} a^{2} + \frac{4}{41} a - \frac{14}{41}$, $\frac{1}{8408336457} a^{16} + \frac{98818639}{8408336457} a^{15} - \frac{67650587}{8408336457} a^{14} + \frac{961516885}{2802778819} a^{13} - \frac{622801396}{8408336457} a^{12} - \frac{64781411}{8408336457} a^{11} - \frac{618398784}{2802778819} a^{10} + \frac{3431181439}{8408336457} a^{9} - \frac{3447711685}{8408336457} a^{8} + \frac{3242626922}{8408336457} a^{7} - \frac{1217687603}{8408336457} a^{6} + \frac{8351492}{8408336457} a^{5} - \frac{579581318}{8408336457} a^{4} + \frac{843624652}{2802778819} a^{3} + \frac{1319012238}{2802778819} a^{2} - \frac{999782551}{8408336457} a - \frac{1277212355}{2802778819}$, $\frac{1}{123592432791880318434616089} a^{17} - \frac{3502832895870416}{123592432791880318434616089} a^{16} + \frac{5902835453302926672142}{1274148791668869262212537} a^{15} - \frac{72784900529358955607082}{13732492532431146492735121} a^{14} + \frac{915076302767128007597089}{3014449580289763864258929} a^{13} - \frac{48780632762166263214880304}{123592432791880318434616089} a^{12} + \frac{1853558767794566839745341}{41197477597293439478205363} a^{11} - \frac{36010615422451199957045849}{123592432791880318434616089} a^{10} + \frac{32776126206012418402488176}{123592432791880318434616089} a^{9} + \frac{41428522126003421086677791}{123592432791880318434616089} a^{8} + \frac{7984772058560866665739036}{123592432791880318434616089} a^{7} - \frac{22120811069404304395080883}{123592432791880318434616089} a^{6} - \frac{31662164058645198367934591}{123592432791880318434616089} a^{5} + \frac{3369221319031760329224386}{13732492532431146492735121} a^{4} + \frac{19816277875561463703716833}{41197477597293439478205363} a^{3} + \frac{54116305906808367499828412}{123592432791880318434616089} a^{2} + \frac{13809842082307502591519536}{41197477597293439478205363} a - \frac{5448074775270510144099077}{13732492532431146492735121}$

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

Class group and class number

$C_{9}$, which has order $9$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 42459.5790611 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3$ (as 18T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 9 conjugacy class representatives for $S_3 \times C_3$
Character table for $S_3 \times C_3$

Intermediate fields

\(\Q(\sqrt{-59}) \), 3.1.2891.1 x3, \(\Q(\zeta_{7})^+\), 6.0.493114979.3, 6.0.10063571.3 x2, 6.0.493114979.4, 9.3.24162633971.1 x3

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

Sibling fields

Degree 6 sibling: 6.0.10063571.3
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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ R

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
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
$59$59.6.3.2$x^{6} - 3481 x^{2} + 3491443$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
59.6.3.2$x^{6} - 3481 x^{2} + 3491443$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
59.6.3.2$x^{6} - 3481 x^{2} + 3491443$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$