↑

Properties

Label 18.0.10293947901...7136.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{9}\cdot 7^{15}\cdot 11^{6}\cdot 19^{15}$
Root discriminant $359.96$
Ramified primes $2, 3, 7, 11, 19$
Class number $4257850752$ (GRH)
Class group $[2, 2, 2, 6, 88705224]$ (GRH)
Galois group $S_3 \times C_6$ (as 18T6)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![22784958920704, 13574001819584, 10112290341964, 3744823323947, 1482883758697, 368692180073, 99487490867, 18385432151, 3990653485, 597868077, 116404855, 13020561, 2169003, 119723, 24585, 237, 191, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 + 191*x^16 + 237*x^15 + 24585*x^14 + 119723*x^13 + 2169003*x^12 + 13020561*x^11 + 116404855*x^10 + 597868077*x^9 + 3990653485*x^8 + 18385432151*x^7 + 99487490867*x^6 + 368692180073*x^5 + 1482883758697*x^4 + 3744823323947*x^3 + 10112290341964*x^2 + 13574001819584*x + 22784958920704)
 
gp: K = bnfinit(x^18 - x^17 + 191*x^16 + 237*x^15 + 24585*x^14 + 119723*x^13 + 2169003*x^12 + 13020561*x^11 + 116404855*x^10 + 597868077*x^9 + 3990653485*x^8 + 18385432151*x^7 + 99487490867*x^6 + 368692180073*x^5 + 1482883758697*x^4 + 3744823323947*x^3 + 10112290341964*x^2 + 13574001819584*x + 22784958920704, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} + 191 x^{16} + 237 x^{15} + 24585 x^{14} + 119723 x^{13} + 2169003 x^{12} + 13020561 x^{11} + 116404855 x^{10} + 597868077 x^{9} + 3990653485 x^{8} + 18385432151 x^{7} + 99487490867 x^{6} + 368692180073 x^{5} + 1482883758697 x^{4} + 3744823323947 x^{3} + 10112290341964 x^{2} + 13574001819584 x + 22784958920704 \)

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:  \(-10293947901026815839844463877644638832922587136=-\,2^{12}\cdot 3^{9}\cdot 7^{15}\cdot 11^{6}\cdot 19^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $359.96$
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:  $2, 3, 7, 11, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{4} a$, $\frac{1}{4} a^{5} - \frac{1}{4} a$, $\frac{1}{8} a^{6} - \frac{1}{8} a^{2}$, $\frac{1}{32} a^{7} + \frac{1}{32} a^{6} - \frac{1}{8} a^{4} + \frac{3}{32} a^{3} + \frac{3}{32} a^{2} - \frac{1}{8} a$, $\frac{1}{32} a^{8} - \frac{1}{32} a^{6} - \frac{1}{8} a^{5} - \frac{1}{32} a^{4} - \frac{1}{4} a^{3} - \frac{15}{32} a^{2} - \frac{1}{8} a$, $\frac{1}{448} a^{9} + \frac{3}{448} a^{8} - \frac{1}{64} a^{7} + \frac{3}{448} a^{6} - \frac{5}{448} a^{5} - \frac{19}{448} a^{4} - \frac{89}{448} a^{3} + \frac{13}{448} a^{2} + \frac{25}{112} a - \frac{1}{7}$, $\frac{1}{2240} a^{10} - \frac{1}{2240} a^{9} - \frac{33}{2240} a^{8} - \frac{11}{2240} a^{7} - \frac{101}{2240} a^{6} + \frac{57}{2240} a^{5} - \frac{167}{2240} a^{4} + \frac{467}{2240} a^{3} + \frac{15}{112} a^{2} - \frac{9}{35} a + \frac{4}{35}$, $\frac{1}{8960} a^{11} - \frac{1}{4480} a^{10} - \frac{1}{4480} a^{9} - \frac{7}{640} a^{8} - \frac{1}{448} a^{7} + \frac{89}{4480} a^{6} + \frac{233}{4480} a^{5} - \frac{143}{4480} a^{4} + \frac{243}{8960} a^{3} - \frac{271}{560} a^{2} - \frac{23}{560} a + \frac{9}{35}$, $\frac{1}{35840} a^{12} - \frac{1}{35840} a^{11} - \frac{1}{4480} a^{10} - \frac{9}{8960} a^{9} - \frac{43}{17920} a^{8} + \frac{31}{17920} a^{7} + \frac{167}{8960} a^{6} - \frac{927}{8960} a^{5} + \frac{177}{7168} a^{4} + \frac{2979}{35840} a^{3} + \frac{117}{1280} a^{2} - \frac{31}{112} a + \frac{3}{7}$, $\frac{1}{143360} a^{13} + \frac{1}{20480} a^{11} - \frac{3}{35840} a^{10} - \frac{29}{71680} a^{9} + \frac{157}{17920} a^{8} + \frac{59}{10240} a^{7} - \frac{263}{4480} a^{6} + \frac{16441}{143360} a^{5} - \frac{249}{3584} a^{4} + \frac{14591}{143360} a^{3} - \frac{439}{1024} a^{2} + \frac{227}{2240} a + \frac{11}{70}$, $\frac{1}{143360} a^{14} - \frac{1}{143360} a^{12} - \frac{1}{35840} a^{11} + \frac{3}{71680} a^{10} - \frac{1}{2560} a^{9} + \frac{597}{71680} a^{8} - \frac{137}{8960} a^{7} - \frac{2663}{143360} a^{6} + \frac{1223}{17920} a^{5} + \frac{15511}{143360} a^{4} - \frac{7003}{35840} a^{3} - \frac{5}{128} a^{2} + \frac{193}{560} a - \frac{1}{7}$, $\frac{1}{573440} a^{15} - \frac{1}{143360} a^{12} - \frac{19}{573440} a^{11} - \frac{1}{143360} a^{10} + \frac{3}{7168} a^{9} + \frac{207}{14336} a^{8} - \frac{1131}{81920} a^{7} + \frac{779}{71680} a^{6} + \frac{103}{1024} a^{5} - \frac{1873}{28672} a^{4} + \frac{9267}{114688} a^{3} + \frac{13103}{28672} a^{2} - \frac{1129}{8960} a - \frac{13}{56}$, $\frac{1}{1358181171200} a^{16} + \frac{132317}{1358181171200} a^{15} + \frac{1165329}{339545292800} a^{14} - \frac{309699}{169772646400} a^{13} + \frac{78661}{12019302400} a^{12} + \frac{14024249}{1358181171200} a^{11} + \frac{9841473}{339545292800} a^{10} - \frac{82342077}{84886323200} a^{9} - \frac{5029009773}{1358181171200} a^{8} - \frac{1700891061}{271636234240} a^{7} + \frac{11998511387}{339545292800} a^{6} - \frac{709803743}{33954529280} a^{5} - \frac{79995670281}{1358181171200} a^{4} + \frac{17853910551}{271636234240} a^{3} - \frac{165829932973}{339545292800} a^{2} + \frac{9948787679}{21221580800} a + \frac{2851151}{5868800}$, $\frac{1}{31324456626910685066296914878284184948765260530750649794560000} a^{17} - \frac{3705438235826284270724803028321189974898069852801}{15662228313455342533148457439142092474382630265375324897280000} a^{16} + \frac{12833167150981146969835825238257649783094432221688748993}{31324456626910685066296914878284184948765260530750649794560000} a^{15} + \frac{8003879645885215112997711149652268310124719948254585611}{7831114156727671266574228719571046237191315132687662448640000} a^{14} - \frac{40886641412990397790239843713374227806737008518971828259}{31324456626910685066296914878284184948765260530750649794560000} a^{13} + \frac{135313851028426980003185896224189369832191346200072953191}{15662228313455342533148457439142092474382630265375324897280000} a^{12} + \frac{142999200347219514481736828478936888686199677922167774203}{4474922375272955009470987839754883564109322932964378542080000} a^{11} - \frac{52725704839047015133229049694517872463535289953974719223}{1566222831345534253314845743914209247438263026537532489728000} a^{10} - \frac{1909337381226601719650443864400976825537451594329453304537}{6264891325382137013259382975656836989753052106150129958912000} a^{9} - \frac{76014391964233577025375137732724436115406525703467788428619}{15662228313455342533148457439142092474382630265375324897280000} a^{8} - \frac{119855021784731316253823941317183168112422843082353950300477}{31324456626910685066296914878284184948765260530750649794560000} a^{7} - \frac{188129144715807674680037526726332077067107407258550926158043}{7831114156727671266574228719571046237191315132687662448640000} a^{6} - \frac{3331898840083221224551336141817811278456092477540757954321761}{31324456626910685066296914878284184948765260530750649794560000} a^{5} + \frac{1774026358579030727232978575348577183816457244496059993039717}{15662228313455342533148457439142092474382630265375324897280000} a^{4} - \frac{7577415546015026129513057535817106489548370740114582633495137}{31324456626910685066296914878284184948765260530750649794560000} a^{3} + \frac{233199211498205489872653046716930213060680629392784214209571}{7831114156727671266574228719571046237191315132687662448640000} a^{2} - \frac{13259507462461466026864055332294177293514358696533285528621}{97888926959095890832177858994638077964891439158595780608000} a - \frac{3353836525118154334357593222629119534278792643877218129}{135355264047422415420599915645240705150568914765757440000}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{6}\times C_{88705224}$, which has order $4257850752$ (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:  \( 2494653063.4840164 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6\times S_3$ (as 18T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 18 conjugacy class representatives for $S_3 \times C_6$
Character table for $S_3 \times C_6$

Intermediate fields

\(\Q(\sqrt{-399}) \), 3.3.17689.2, 3.3.15884.1, 6.0.1123626489111.2, 6.0.44394711896304.1, 9.9.471484994327458496.3

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

Sibling fields

Galois closure: data not computed
Degree 12 sibling: data not computed
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 ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R R ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
$3$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
7Data not computed
$11$11.6.0.1$x^{6} + x^{2} - 2 x + 8$$1$$6$$0$$C_6$$[\ ]^{6}$
11.12.6.1$x^{12} + 242 x^{8} + 21296 x^{6} + 14641 x^{4} + 1932612 x^{2} + 113379904$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$19$19.6.5.3$x^{6} - 4864$$6$$1$$5$$C_6$$[\ ]_{6}$
19.6.5.3$x^{6} - 4864$$6$$1$$5$$C_6$$[\ ]_{6}$
19.6.5.3$x^{6} - 4864$$6$$1$$5$$C_6$$[\ ]_{6}$