↑

Properties

Label 18.0.20138449584...6688.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{33}\cdot 19^{12}\cdot 43^{12}$
Root discriminant $1039.66$
Ramified primes $2, 3, 19, 43$
Class number $79823146788$ (GRH)
Class group $[3, 9, 9, 9, 1386, 26334]$ (GRH)
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![4995175830121, 433013473827, 462523978377, -293292196752, 11484916434, -18851833098, 7075342842, -590965020, 237549663, -57195115, 2184723, -57708, -29010, 21690, -5526, 636, 45, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 45*x^16 + 636*x^15 - 5526*x^14 + 21690*x^13 - 29010*x^12 - 57708*x^11 + 2184723*x^10 - 57195115*x^9 + 237549663*x^8 - 590965020*x^7 + 7075342842*x^6 - 18851833098*x^5 + 11484916434*x^4 - 293292196752*x^3 + 462523978377*x^2 + 433013473827*x + 4995175830121)
 
gp: K = bnfinit(x^18 - 9*x^17 + 45*x^16 + 636*x^15 - 5526*x^14 + 21690*x^13 - 29010*x^12 - 57708*x^11 + 2184723*x^10 - 57195115*x^9 + 237549663*x^8 - 590965020*x^7 + 7075342842*x^6 - 18851833098*x^5 + 11484916434*x^4 - 293292196752*x^3 + 462523978377*x^2 + 433013473827*x + 4995175830121, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 45 x^{16} + 636 x^{15} - 5526 x^{14} + 21690 x^{13} - 29010 x^{12} - 57708 x^{11} + 2184723 x^{10} - 57195115 x^{9} + 237549663 x^{8} - 590965020 x^{7} + 7075342842 x^{6} - 18851833098 x^{5} + 11484916434 x^{4} - 293292196752 x^{3} + 462523978377 x^{2} + 433013473827 x + 4995175830121 \)

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:  \(-2013844958403898050255539081540690226371968347203186688=-\,2^{12}\cdot 3^{33}\cdot 19^{12}\cdot 43^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1039.66$
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:  $2, 3, 19, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{6} a^{4} - \frac{1}{2} a^{2} - \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{6} a^{5} - \frac{1}{6} a^{3} - \frac{1}{3} a^{2} - \frac{1}{2} a + \frac{1}{3}$, $\frac{1}{36} a^{6} - \frac{1}{12} a^{5} + \frac{5}{36} a^{3} - \frac{1}{3} a^{2} + \frac{1}{4} a - \frac{5}{36}$, $\frac{1}{36} a^{7} - \frac{1}{12} a^{5} - \frac{1}{36} a^{4} - \frac{1}{12} a^{3} + \frac{5}{12} a^{2} + \frac{4}{9} a + \frac{5}{12}$, $\frac{1}{36} a^{8} + \frac{1}{18} a^{5} - \frac{1}{12} a^{4} - \frac{1}{6} a^{3} - \frac{2}{9} a^{2} + \frac{1}{6} a - \frac{5}{12}$, $\frac{1}{108} a^{9} - \frac{1}{12} a^{5} + \frac{1}{18} a^{3} - \frac{1}{3} a^{2} + \frac{1}{4} a - \frac{5}{27}$, $\frac{1}{216} a^{10} - \frac{1}{216} a^{9} - \frac{1}{72} a^{8} - \frac{1}{72} a^{6} + \frac{1}{72} a^{5} - \frac{1}{72} a^{4} - \frac{1}{18} a^{3} + \frac{11}{72} a^{2} - \frac{29}{216} a + \frac{17}{216}$, $\frac{1}{648} a^{11} - \frac{1}{648} a^{10} + \frac{1}{648} a^{9} - \frac{1}{108} a^{8} - \frac{1}{216} a^{7} + \frac{1}{216} a^{6} - \frac{17}{216} a^{5} + \frac{1}{108} a^{4} + \frac{7}{216} a^{3} + \frac{91}{648} a^{2} - \frac{127}{648} a + \frac{77}{324}$, $\frac{1}{1296} a^{12} - \frac{1}{432} a^{10} - \frac{1}{648} a^{9} - \frac{1}{432} a^{6} - \frac{1}{24} a^{5} - \frac{1}{18} a^{4} - \frac{53}{324} a^{3} - \frac{17}{48} a^{2} + \frac{55}{216} a - \frac{617}{1296}$, $\frac{1}{1296} a^{13} - \frac{1}{1296} a^{11} + \frac{1}{648} a^{10} - \frac{1}{324} a^{9} + \frac{1}{216} a^{8} - \frac{1}{144} a^{7} + \frac{1}{216} a^{6} - \frac{7}{108} a^{5} + \frac{53}{648} a^{4} - \frac{43}{432} a^{3} + \frac{211}{648} a^{2} - \frac{397}{1296} a + \frac{187}{648}$, $\frac{1}{42768} a^{14} - \frac{7}{42768} a^{13} - \frac{5}{42768} a^{12} - \frac{1}{3888} a^{11} - \frac{1}{10692} a^{10} + \frac{4}{2673} a^{9} + \frac{29}{14256} a^{8} + \frac{31}{14256} a^{7} - \frac{7}{3564} a^{6} + \frac{829}{10692} a^{5} + \frac{2087}{42768} a^{4} + \frac{6937}{42768} a^{3} + \frac{16987}{42768} a^{2} - \frac{11773}{42768} a + \frac{5215}{10692}$, $\frac{1}{7304723463312} a^{15} + \frac{2030753}{202908985092} a^{14} + \frac{1943345}{27669407058} a^{13} + \frac{108367711}{3652361731656} a^{12} + \frac{1157857949}{2434907821104} a^{11} + \frac{117104354}{50727246273} a^{10} + \frac{6948030355}{7304723463312} a^{9} + \frac{445301663}{202908985092} a^{8} + \frac{4908075215}{811635940368} a^{7} + \frac{11737503089}{913090432914} a^{6} + \frac{22539435355}{811635940368} a^{5} + \frac{1765265197}{304363477638} a^{4} + \frac{6771983125}{117818120376} a^{3} - \frac{1643868893}{16019130402} a^{2} + \frac{1873703515}{14239227024} a - \frac{15017497813}{192229564824}$, $\frac{1}{3053374407664416} a^{16} + \frac{1}{24623987158584} a^{15} - \frac{2953395607}{254447867305368} a^{14} - \frac{39433185193}{763343601916104} a^{13} - \frac{204702240269}{1526687203832208} a^{12} + \frac{1701071209}{254447867305368} a^{11} + \frac{690215297191}{763343601916104} a^{10} + \frac{2054461750441}{763343601916104} a^{9} + \frac{410178825181}{113087941024608} a^{8} + \frac{2319274267603}{763343601916104} a^{7} - \frac{347097408593}{763343601916104} a^{6} + \frac{229686017141}{8207995719528} a^{5} - \frac{28812153434381}{1526687203832208} a^{4} - \frac{611722592341}{10043994762054} a^{3} + \frac{2067357257165}{6695996508036} a^{2} + \frac{8676275382575}{20087989524108} a - \frac{3268304776871}{8458100852256}$, $\frac{1}{50125468460367923842726755261280007136} a^{17} + \frac{585493105615824554545}{4556860769124356712975159569207273376} a^{16} + \frac{374536578368134532001271}{12531367115091980960681688815320001784} a^{15} - \frac{5720321123565850356935546103901}{2278430384562178356487579784603636688} a^{14} + \frac{1526727572218636277915964777630559}{12531367115091980960681688815320001784} a^{13} + \frac{455818825597021937138697150219247}{25062734230183961921363377630640003568} a^{12} + \frac{6765537956670925988000295278699921}{25062734230183961921363377630640003568} a^{11} - \frac{23601590940979235796073434608987}{25062734230183961921363377630640003568} a^{10} + \frac{208309205844988619294636253349801363}{50125468460367923842726755261280007136} a^{9} - \frac{390488293187973362948252532507435113}{50125468460367923842726755261280007136} a^{8} + \frac{9570732002871327389062806465835991}{2278430384562178356487579784603636688} a^{7} + \frac{184024714688835726823945762742890145}{25062734230183961921363377630640003568} a^{6} + \frac{1959302625459159735926734027435620349}{25062734230183961921363377630640003568} a^{5} + \frac{78112077609037286869182238424949053}{1139215192281089178243789892301818344} a^{4} - \frac{161653117411034773729365587203643507}{1319091275272840101124388296349473872} a^{3} - \frac{113127693602131113772868429670222781}{329772818818210025281097074087368468} a^{2} - \frac{760353700499169981381165159509519875}{2638182550545680202248776592698947744} a - \frac{515693692976720628221880816185}{1180400686779970819654493419296}$

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

Class group and class number

$C_{3}\times C_{9}\times C_{9}\times C_{9}\times C_{1386}\times C_{26334}$, which has order $79823146788$ (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:  \( -\frac{197848916237127894406}{1438403020556930780610845823613407} a^{17} + \frac{3311837824315403697019}{1438403020556930780610845823613407} a^{16} + \frac{1937141911246052511064}{1438403020556930780610845823613407} a^{15} - \frac{240925959939726802688192}{1438403020556930780610845823613407} a^{14} + \frac{2848804106161396153976516}{1438403020556930780610845823613407} a^{13} + \frac{192883573746744716716316}{46400097437320347761640187858497} a^{12} - \frac{81767286897111573791014432}{1438403020556930780610845823613407} a^{11} + \frac{333600056494639393060587568}{1438403020556930780610845823613407} a^{10} + \frac{37610742375087109045006810}{75705422134575304242676095979653} a^{9} + \frac{6685478763227106543386431346}{1438403020556930780610845823613407} a^{8} - \frac{68770836487517008989315427616}{1438403020556930780610845823613407} a^{7} - \frac{744575486640286582885497069536}{1438403020556930780610845823613407} a^{6} + \frac{57531059876440652004062140780}{75705422134575304242676095979653} a^{5} + \frac{12578107221189351594213304254104}{1438403020556930780610845823613407} a^{4} + \frac{2810386642329331902738083080904}{75705422134575304242676095979653} a^{3} - \frac{67500962245849730297355098128}{3984495901819752854877689262087} a^{2} - \frac{39591099235784465155286266447202}{75705422134575304242676095979653} a - \frac{19821696153252022940648558}{33872838807965186514419577} \) (order $6$)
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:  \( 2105795997.736377 \) (assuming GRH)
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

\(\Q(\sqrt{-3}) \), 3.1.648799308.1 x3, 3.3.667489.2, 6.0.1262821626183836592.1, 6.0.12029622258267.1, 6.0.1891898782128.3 x2, Deg 9 x3

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

Sibling fields

Degree 6 sibling: data not computed
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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
2Data not computed
3Data not computed
$19$19.3.2.3$x^{3} - 304$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.3$x^{3} - 304$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.3$x^{3} - 304$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.3$x^{3} - 304$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.3$x^{3} - 304$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.3$x^{3} - 304$$3$$1$$2$$C_3$$[\ ]_{3}$
$43$43.9.6.1$x^{9} + 1290 x^{6} + 552851 x^{3} + 79507000$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
43.9.6.1$x^{9} + 1290 x^{6} + 552851 x^{3} + 79507000$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$