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Properties

Label 18.0.13944985186...3231.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{9}\cdot 7^{12}\cdot 13^{15}$
Root discriminant $53.73$
Ramified primes $3, 7, 13$
Class number $1216$ (GRH)
Class group $[2, 2, 2, 2, 76]$ (GRH)
Galois group $C_6 \times C_3$ (as 18T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![215851, -11584, 162369, -115871, 224883, -112288, 69570, -33440, 54562, -15190, 2538, -344, -184, -22, 49, -14, 13, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 13*x^16 - 14*x^15 + 49*x^14 - 22*x^13 - 184*x^12 - 344*x^11 + 2538*x^10 - 15190*x^9 + 54562*x^8 - 33440*x^7 + 69570*x^6 - 112288*x^5 + 224883*x^4 - 115871*x^3 + 162369*x^2 - 11584*x + 215851)
 
gp: K = bnfinit(x^18 - 3*x^17 + 13*x^16 - 14*x^15 + 49*x^14 - 22*x^13 - 184*x^12 - 344*x^11 + 2538*x^10 - 15190*x^9 + 54562*x^8 - 33440*x^7 + 69570*x^6 - 112288*x^5 + 224883*x^4 - 115871*x^3 + 162369*x^2 - 11584*x + 215851, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 13 x^{16} - 14 x^{15} + 49 x^{14} - 22 x^{13} - 184 x^{12} - 344 x^{11} + 2538 x^{10} - 15190 x^{9} + 54562 x^{8} - 33440 x^{7} + 69570 x^{6} - 112288 x^{5} + 224883 x^{4} - 115871 x^{3} + 162369 x^{2} - 11584 x + 215851 \)

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:  \(-13944985186220076513047292273231=-\,3^{9}\cdot 7^{12}\cdot 13^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $53.73$
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:  $3, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(273=3\cdot 7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{273}(256,·)$, $\chi_{273}(1,·)$, $\chi_{273}(134,·)$, $\chi_{273}(79,·)$, $\chi_{273}(16,·)$, $\chi_{273}(211,·)$, $\chi_{273}(212,·)$, $\chi_{273}(22,·)$, $\chi_{273}(23,·)$, $\chi_{273}(218,·)$, $\chi_{273}(155,·)$, $\chi_{273}(95,·)$, $\chi_{273}(100,·)$, $\chi_{273}(233,·)$, $\chi_{273}(235,·)$, $\chi_{273}(172,·)$, $\chi_{273}(179,·)$, $\chi_{273}(116,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4}$, $\frac{1}{30} a^{12} - \frac{1}{6} a^{11} - \frac{1}{30} a^{10} - \frac{1}{5} a^{9} - \frac{7}{30} a^{8} + \frac{7}{30} a^{7} - \frac{2}{5} a^{6} - \frac{1}{6} a^{5} + \frac{13}{30} a^{4} - \frac{13}{30} a^{3} - \frac{1}{6} a + \frac{7}{30}$, $\frac{1}{60} a^{13} - \frac{1}{60} a^{12} + \frac{3}{20} a^{11} + \frac{1}{12} a^{10} - \frac{1}{60} a^{9} + \frac{3}{20} a^{8} + \frac{1}{60} a^{7} + \frac{7}{60} a^{6} - \frac{7}{60} a^{5} - \frac{7}{20} a^{4} + \frac{23}{60} a^{3} + \frac{5}{12} a^{2} - \frac{13}{60} a - \frac{17}{60}$, $\frac{1}{60} a^{14} - \frac{1}{10} a^{11} + \frac{1}{5} a^{10} - \frac{1}{15} a^{9} + \frac{1}{10} a^{8} + \frac{1}{5} a^{7} - \frac{2}{5} a^{6} + \frac{1}{5} a^{5} + \frac{3}{10} a^{4} - \frac{7}{15} a^{3} + \frac{1}{5} a^{2} + \frac{1}{6} a - \frac{13}{60}$, $\frac{1}{300} a^{15} - \frac{1}{150} a^{14} - \frac{1}{150} a^{13} + \frac{1}{75} a^{12} - \frac{1}{75} a^{11} + \frac{11}{75} a^{10} + \frac{7}{75} a^{9} + \frac{23}{150} a^{8} + \frac{11}{50} a^{7} + \frac{13}{30} a^{6} + \frac{7}{75} a^{5} - \frac{32}{75} a^{4} - \frac{13}{75} a^{3} - \frac{1}{75} a^{2} - \frac{47}{300} a - \frac{16}{75}$, $\frac{1}{300} a^{16} - \frac{1}{300} a^{14} + \frac{1}{75} a^{12} + \frac{1}{50} a^{11} + \frac{13}{150} a^{10} - \frac{17}{75} a^{9} + \frac{19}{150} a^{8} + \frac{11}{150} a^{7} - \frac{11}{25} a^{6} - \frac{1}{25} a^{5} + \frac{41}{150} a^{4} - \frac{49}{150} a^{3} - \frac{29}{60} a^{2} + \frac{7}{50} a + \frac{107}{300}$, $\frac{1}{896882960878596456291765410917951037245200} a^{17} + \frac{442178448292571750278966898709829369223}{448441480439298228145882705458975518622600} a^{16} + \frac{469741660810163608221285344584642546121}{298960986959532152097255136972650345748400} a^{15} - \frac{3383079541025426216716977486375741754039}{896882960878596456291765410917951037245200} a^{14} + \frac{240977676030709182866116018127308992863}{448441480439298228145882705458975518622600} a^{13} + \frac{11415380664351664677804326355147537092}{18685061684970759506078446060790646609275} a^{12} - \frac{20062099377376814752238110487073811914733}{112110370109824557036470676364743879655650} a^{11} + \frac{4423760516300632921968828766195076442131}{37370123369941519012156892121581293218550} a^{10} + \frac{8715192804142967715702724563030224463007}{149480493479766076048627568486325172874200} a^{9} + \frac{35021326488072248516130530346466026190021}{224220740219649114072941352729487759311300} a^{8} + \frac{13984324426886417216042959298885537241169}{149480493479766076048627568486325172874200} a^{7} - \frac{3187546526625190810626050413401940761099}{149480493479766076048627568486325172874200} a^{6} - \frac{142040495969630747860063116570738236417}{37370123369941519012156892121581293218550} a^{5} - \frac{7807296204320892443509784440430938930797}{37370123369941519012156892121581293218550} a^{4} - \frac{145577023538590008624940950558813200153021}{896882960878596456291765410917951037245200} a^{3} + \frac{309471358295658258775343824531238242253}{74740246739883038024313784243162586437100} a^{2} + \frac{81774648422546615477349915718240574298927}{298960986959532152097255136972650345748400} a - \frac{327250677181055400127523749525880968764239}{896882960878596456291765410917951037245200}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{76}$, which has order $1216$ (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:  \( 205236.825908 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_6$ (as 18T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 18
The 18 conjugacy class representatives for $C_6 \times C_3$
Character table for $C_6 \times C_3$

Intermediate fields

\(\Q(\sqrt{-39}) \), 3.3.8281.1, 3.3.169.1, \(\Q(\zeta_{7})^+\), 3.3.8281.2, 6.0.24069811311.1, 6.0.10024911.1, 6.0.142424919.1, 6.0.24069811311.2, 9.9.567869252041.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$3$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7Data not computed
$13$13.6.5.2$x^{6} - 13$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.5.2$x^{6} - 13$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.5.2$x^{6} - 13$$6$$1$$5$$C_6$$[\ ]_{6}$