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Properties

Label 18.0.98752947213...0907.4
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{39}\cdot 7^{12}\cdot 127^{12}$
Root discriminant $999.30$
Ramified primes $3, 7, 127$
Class number $19999187712$ (GRH)
Class group $[3, 18, 18, 18, 18, 252, 252]$ (GRH)
Galois group $S_3 \times C_3$ (as 18T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![780420096, 396747072, 573552900, -352468422, 91894095, -103026411, 49427415, -12557430, 5627718, -1489446, 90828, 26028, -12816, 4590, -1026, 36, 45, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 45*x^16 + 36*x^15 - 1026*x^14 + 4590*x^13 - 12816*x^12 + 26028*x^11 + 90828*x^10 - 1489446*x^9 + 5627718*x^8 - 12557430*x^7 + 49427415*x^6 - 103026411*x^5 + 91894095*x^4 - 352468422*x^3 + 573552900*x^2 + 396747072*x + 780420096)
 
gp: K = bnfinit(x^18 - 9*x^17 + 45*x^16 + 36*x^15 - 1026*x^14 + 4590*x^13 - 12816*x^12 + 26028*x^11 + 90828*x^10 - 1489446*x^9 + 5627718*x^8 - 12557430*x^7 + 49427415*x^6 - 103026411*x^5 + 91894095*x^4 - 352468422*x^3 + 573552900*x^2 + 396747072*x + 780420096, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 45 x^{16} + 36 x^{15} - 1026 x^{14} + 4590 x^{13} - 12816 x^{12} + 26028 x^{11} + 90828 x^{10} - 1489446 x^{9} + 5627718 x^{8} - 12557430 x^{7} + 49427415 x^{6} - 103026411 x^{5} + 91894095 x^{4} - 352468422 x^{3} + 573552900 x^{2} + 396747072 x + 780420096 \)

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:  \(-987529472131924046848116539449875888434135640677420907=-\,3^{39}\cdot 7^{12}\cdot 127^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $999.30$
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:  $3, 7, 127$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{6} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{6} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{6} a^{7} - \frac{1}{2} a$, $\frac{1}{12} a^{8} + \frac{1}{4} a^{2}$, $\frac{1}{108} a^{9} - \frac{1}{12} a^{3} - \frac{1}{3}$, $\frac{1}{108} a^{10} - \frac{1}{12} a^{4} - \frac{1}{3} a$, $\frac{1}{324} a^{11} - \frac{1}{36} a^{8} - \frac{1}{36} a^{5} + \frac{1}{6} a^{4} - \frac{1}{3} a^{3} + \frac{17}{36} a^{2} - \frac{1}{2} a$, $\frac{1}{57024} a^{12} - \frac{1}{9504} a^{11} + \frac{1}{1056} a^{10} + \frac{1}{6336} a^{9} - \frac{17}{1056} a^{8} - \frac{2}{33} a^{7} + \frac{1}{576} a^{6} - \frac{13}{1056} a^{5} - \frac{107}{1056} a^{4} - \frac{325}{6336} a^{3} - \frac{233}{1056} a^{2} - \frac{9}{176} a + \frac{4}{11}$, $\frac{1}{57024} a^{13} + \frac{1}{3168} a^{11} - \frac{65}{19008} a^{10} + \frac{1}{297} a^{9} + \frac{5}{528} a^{8} - \frac{181}{6336} a^{7} - \frac{1}{528} a^{6} - \frac{3}{352} a^{5} - \frac{481}{6336} a^{4} + \frac{161}{528} a^{3} - \frac{3}{8} a^{2} - \frac{29}{264} a - \frac{16}{33}$, $\frac{1}{114048} a^{14} - \frac{1}{114048} a^{13} + \frac{71}{114048} a^{11} - \frac{19}{38016} a^{10} + \frac{31}{19008} a^{9} - \frac{5}{384} a^{8} - \frac{311}{12672} a^{7} + \frac{17}{264} a^{6} - \frac{85}{4224} a^{5} - \frac{1343}{12672} a^{4} + \frac{433}{2112} a^{3} + \frac{727}{3168} a^{2} - \frac{19}{132} a - \frac{1}{33}$, $\frac{1}{16817404032} a^{15} - \frac{3583}{2802900672} a^{14} + \frac{1003}{5605801344} a^{13} + \frac{23713}{5605801344} a^{12} + \frac{875}{589836} a^{11} + \frac{4024963}{1868600448} a^{10} - \frac{1611875}{1868600448} a^{9} - \frac{5293147}{311433408} a^{8} + \frac{4490029}{622866816} a^{7} - \frac{11970929}{169872768} a^{6} - \frac{133067}{3539016} a^{5} + \frac{17028335}{622866816} a^{4} - \frac{4662107}{16391232} a^{3} + \frac{5693477}{17301856} a^{2} - \frac{4802665}{12976392} a + \frac{410804}{1622049}$, $\frac{1}{538156929024} a^{16} - \frac{1}{67269616128} a^{15} + \frac{1}{14948803584} a^{14} - \frac{98305}{11211602688} a^{13} - \frac{223987}{29897607168} a^{12} + \frac{1794307}{22423205376} a^{11} + \frac{7005079}{3321956352} a^{10} + \frac{14336329}{3321956352} a^{9} - \frac{47874439}{1245733632} a^{8} - \frac{572014565}{7474401792} a^{7} - \frac{99325637}{2299815936} a^{6} + \frac{33515327}{622866816} a^{5} + \frac{4209543787}{19931738112} a^{4} + \frac{1421500943}{3321956352} a^{3} + \frac{2424978019}{4982934528} a^{2} - \frac{3470213}{9437376} a - \frac{1633807}{6488196}$, $\frac{1}{6300791010927830211784704} a^{17} - \frac{1857534248173}{2100263670309276737261568} a^{16} + \frac{15060844897405}{525065917577319184315392} a^{15} + \frac{2033909520154599479}{525065917577319184315392} a^{14} - \frac{67651280447142709}{80779371934972182202368} a^{13} - \frac{3388821739417972237}{1050131835154638368630784} a^{12} + \frac{391180315009582427401}{1050131835154638368630784} a^{11} - \frac{75233537566501407389}{21877746565721632679808} a^{10} + \frac{61523697539887557691}{31822176822867829352448} a^{9} - \frac{2131975041327734299331}{87510986262886530719232} a^{8} + \frac{121151414155939875215}{2991828590184154896384} a^{7} + \frac{7935376083195839668229}{116681315017182040958976} a^{6} - \frac{17546211532105203166165}{233362630034364081917952} a^{5} + \frac{3447984867631459288975}{21214784548578552901632} a^{4} - \frac{22463902137580580138765}{116681315017182040958976} a^{3} + \frac{28365484594304696803795}{58340657508591020479488} a^{2} + \frac{330126535477415041693}{1215430364762312926656} a + \frac{37301928130004165}{783138121625201628}$

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

Class group and class number

$C_{3}\times C_{18}\times C_{18}\times C_{18}\times C_{18}\times C_{252}\times C_{252}$, which has order $19999187712$ (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:  \( -\frac{2031058682989}{2237496807857894251344} a^{17} + \frac{27454556352611}{2983329077143859001792} a^{16} - \frac{33847261258789}{745832269285964750448} a^{15} - \frac{75972028926133}{2983329077143859001792} a^{14} + \frac{1144283265074357}{994443025714619667264} a^{13} - \frac{4809361487143709}{994443025714619667264} a^{12} + \frac{11122498494539465}{994443025714619667264} a^{11} - \frac{16794348753807607}{994443025714619667264} a^{10} - \frac{92627769481332451}{994443025714619667264} a^{9} + \frac{1454800864919228761}{994443025714619667264} a^{8} - \frac{1962698936241092567}{331481008571539889088} a^{7} + \frac{3391034488759509979}{331481008571539889088} a^{6} - \frac{925265648048415887}{25498539120887683776} a^{5} + \frac{6387997249074222253}{55246834761923314848} a^{4} - \frac{2961925198451285231}{36831223174615543232} a^{3} + \frac{15681051648521509417}{55246834761923314848} a^{2} - \frac{23779371075086372921}{27623417380961657424} a + \frac{5681127277202248}{17798593673300037} \) (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:  \( 59097926601.30937 \) (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

\(\Q(\sqrt{-3}) \), 3.1.192048003.1 x3, 3.3.64016001.4, 6.0.110647306368864027.1, 6.0.140002994187.1 x2, 6.0.12294145152096003.1, 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
3Data not computed
$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}$
$127$127.3.2.1$x^{3} - 127$$3$$1$$2$$C_3$$[\ ]_{3}$
127.3.2.1$x^{3} - 127$$3$$1$$2$$C_3$$[\ ]_{3}$
127.3.2.1$x^{3} - 127$$3$$1$$2$$C_3$$[\ ]_{3}$
127.3.2.1$x^{3} - 127$$3$$1$$2$$C_3$$[\ ]_{3}$
127.3.2.1$x^{3} - 127$$3$$1$$2$$C_3$$[\ ]_{3}$
127.3.2.1$x^{3} - 127$$3$$1$$2$$C_3$$[\ ]_{3}$