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Properties

Label 18.12.1738227155...7488.1
Degree $18$
Signature $[12, 3]$
Discriminant $-\,2^{12}\cdot 7^{12}\cdot 37^{12}\cdot 167^{3}$
Root discriminant $151.36$
Ramified primes $2, 7, 37, 167$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 18T176

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![101785794048, 33272275968, -68178001536, 22714675904, -3145932384, -119867456, 419590832, -175203836, 47970630, -9914269, -2497471, 1794941, -122314, -70026, 8769, 1089, -165, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 - 165*x^16 + 1089*x^15 + 8769*x^14 - 70026*x^13 - 122314*x^12 + 1794941*x^11 - 2497471*x^10 - 9914269*x^9 + 47970630*x^8 - 175203836*x^7 + 419590832*x^6 - 119867456*x^5 - 3145932384*x^4 + 22714675904*x^3 - 68178001536*x^2 + 33272275968*x + 101785794048)
 
gp: K = bnfinit(x^18 - 6*x^17 - 165*x^16 + 1089*x^15 + 8769*x^14 - 70026*x^13 - 122314*x^12 + 1794941*x^11 - 2497471*x^10 - 9914269*x^9 + 47970630*x^8 - 175203836*x^7 + 419590832*x^6 - 119867456*x^5 - 3145932384*x^4 + 22714675904*x^3 - 68178001536*x^2 + 33272275968*x + 101785794048, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} - 165 x^{16} + 1089 x^{15} + 8769 x^{14} - 70026 x^{13} - 122314 x^{12} + 1794941 x^{11} - 2497471 x^{10} - 9914269 x^{9} + 47970630 x^{8} - 175203836 x^{7} + 419590832 x^{6} - 119867456 x^{5} - 3145932384 x^{4} + 22714675904 x^{3} - 68178001536 x^{2} + 33272275968 x + 101785794048 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-1738227155372083140288705823158000447488=-\,2^{12}\cdot 7^{12}\cdot 37^{12}\cdot 167^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $151.36$
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:  $2, 7, 37, 167$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{12} a^{11} + \frac{1}{4} a^{9} + \frac{1}{4} a^{8} - \frac{1}{12} a^{7} - \frac{1}{3} a^{6} + \frac{1}{6} a^{5} + \frac{5}{12} a^{4} + \frac{1}{4} a^{3} + \frac{1}{12} a^{2} - \frac{1}{3} a$, $\frac{1}{24} a^{12} + \frac{1}{8} a^{10} + \frac{1}{8} a^{9} + \frac{11}{24} a^{8} + \frac{1}{3} a^{7} - \frac{5}{12} a^{6} - \frac{7}{24} a^{5} - \frac{3}{8} a^{4} - \frac{11}{24} a^{3} - \frac{1}{6} a^{2} - \frac{1}{2} a$, $\frac{1}{48} a^{13} - \frac{1}{48} a^{11} + \frac{1}{16} a^{10} - \frac{1}{48} a^{9} + \frac{5}{12} a^{8} + \frac{3}{8} a^{7} - \frac{5}{16} a^{6} - \frac{17}{48} a^{5} + \frac{17}{48} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{6} a$, $\frac{1}{96} a^{14} - \frac{1}{96} a^{12} + \frac{1}{32} a^{11} - \frac{1}{96} a^{10} + \frac{5}{24} a^{9} + \frac{3}{16} a^{8} + \frac{11}{32} a^{7} + \frac{31}{96} a^{6} + \frac{17}{96} a^{5} - \frac{1}{6} a^{4} - \frac{1}{6} a^{3} + \frac{5}{12} a^{2}$, $\frac{1}{192} a^{15} - \frac{1}{192} a^{13} + \frac{1}{64} a^{12} - \frac{1}{192} a^{11} + \frac{5}{48} a^{10} + \frac{3}{32} a^{9} + \frac{11}{64} a^{8} + \frac{31}{192} a^{7} + \frac{17}{192} a^{6} - \frac{1}{12} a^{5} + \frac{5}{12} a^{4} - \frac{7}{24} a^{3}$, $\frac{1}{14976} a^{16} - \frac{1}{468} a^{15} + \frac{7}{14976} a^{14} - \frac{125}{14976} a^{13} - \frac{89}{14976} a^{12} - \frac{125}{3744} a^{11} + \frac{463}{2496} a^{10} - \frac{523}{1152} a^{9} - \frac{7073}{14976} a^{8} + \frac{465}{1664} a^{7} + \frac{187}{624} a^{6} + \frac{833}{1872} a^{5} + \frac{29}{624} a^{4} + \frac{283}{936} a^{3} + \frac{95}{468} a^{2} + \frac{1}{78} a - \frac{5}{13}$, $\frac{1}{157567963256345693714452092029942183002715753703315227080624509423451446528} a^{17} - \frac{669330694375280291736562048617846097902594226779909632739215888286529}{78783981628172846857226046014971091501357876851657613540312254711725723264} a^{16} - \frac{15634231664454494714946401560627253847768848970924499485277001176016849}{12120612558180437978034776309995552538670442592562709775432654571034726656} a^{15} - \frac{559232121813457908283982943458592096821578691005379662804236626052007691}{157567963256345693714452092029942183002715753703315227080624509423451446528} a^{14} - \frac{188299094837408875366565730242106273171756081682379522808347019988690907}{157567963256345693714452092029942183002715753703315227080624509423451446528} a^{13} + \frac{1237457512600468858842975708285517767386053946782813215212280019546455929}{78783981628172846857226046014971091501357876851657613540312254711725723264} a^{12} + \frac{385300022664562593354771224089612417285906170269320025803612876608524737}{26261327209390948952408682004990363833785958950552537846770751570575241088} a^{11} + \frac{13239398503269532949294065522625995250181317805145118431900592941885747373}{157567963256345693714452092029942183002715753703315227080624509423451446528} a^{10} + \frac{57149506677805851801190642460672797280776549947742852451201444196563942005}{157567963256345693714452092029942183002715753703315227080624509423451446528} a^{9} + \frac{69887840044087398752562383724533605545055065639880309152819254827959739}{1346734728686715330892752923332839170963382510284745530603628285670525184} a^{8} + \frac{1809434541786057708829406819162843559963947695616976403303318643936285023}{26261327209390948952408682004990363833785958950552537846770751570575241088} a^{7} + \frac{13379341734024892849471346339520024151050609668728974924976623929741569549}{39391990814086423428613023007485545750678938425828806770156127355862861632} a^{6} + \frac{2345490222071316438759711536461348579436911216399646057661890248328747215}{6565331802347737238102170501247590958446489737638134461692687892643810272} a^{5} - \frac{431528959588424124524884465178479980734649753935823980916173723631966553}{4923998851760802928576627875935693218834867303228600846269515919482857704} a^{4} - \frac{582040373674775981179930179203416094601482291997807264937377034458775235}{2461999425880401464288313937967846609417433651614300423134757959741428852} a^{3} + \frac{7779733136811644824512526707585694762435524353857715248639652089949283}{273555491764489051587590437551982956601937072401588935903861995526825428} a^{2} + \frac{11864627693273529618607562302587070696159170800268899922152805542524994}{205166618823366788690692828163987217451452804301191701927896496645119071} a - \frac{31974640111084849814510319780729177636441362311241484403308593863128060}{68388872941122262896897609387995739150484268100397233975965498881706357}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $14$
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:  \( 13095379124000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T176:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 576
The 40 conjugacy class representatives for t18n176
Character table for t18n176 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 3.3.148.1, 9.9.381393587008.1

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

Sibling fields

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 ${\href{/LocalNumberField/3.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
$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}$
$37$37.6.3.2$x^{6} - 1369 x^{2} + 101306$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
37.12.9.2$x^{12} - 74 x^{8} + 1369 x^{4} - 202612$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
$167$$\Q_{167}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{167}$$x + 2$$1$$1$$0$Trivial$[\ ]$
167.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
167.2.1.2$x^{2} + 334$$2$$1$$1$$C_2$$[\ ]_{2}$
167.4.2.2$x^{4} - 167 x^{2} + 139445$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
167.4.0.1$x^{4} - x + 60$$1$$4$$0$$C_4$$[\ ]^{4}$
167.4.0.1$x^{4} - x + 60$$1$$4$$0$$C_4$$[\ ]^{4}$