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Properties

Label 18.14.8210596569...7297.1
Degree $18$
Signature $[14, 2]$
Discriminant $3^{24}\cdot 73^{6}\cdot 577^{3}$
Root discriminant $52.18$
Ramified primes $3, 73, 577$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T765

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-83079, 374706, 1754109, 1846080, -20277, -1245204, -795618, 31032, 271530, 116072, -11313, -26028, -6386, 1995, 993, -50, -54, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 54*x^16 - 50*x^15 + 993*x^14 + 1995*x^13 - 6386*x^12 - 26028*x^11 - 11313*x^10 + 116072*x^9 + 271530*x^8 + 31032*x^7 - 795618*x^6 - 1245204*x^5 - 20277*x^4 + 1846080*x^3 + 1754109*x^2 + 374706*x - 83079)
 
gp: K = bnfinit(x^18 - 54*x^16 - 50*x^15 + 993*x^14 + 1995*x^13 - 6386*x^12 - 26028*x^11 - 11313*x^10 + 116072*x^9 + 271530*x^8 + 31032*x^7 - 795618*x^6 - 1245204*x^5 - 20277*x^4 + 1846080*x^3 + 1754109*x^2 + 374706*x - 83079, 1)
 

Normalized defining polynomial

\( x^{18} - 54 x^{16} - 50 x^{15} + 993 x^{14} + 1995 x^{13} - 6386 x^{12} - 26028 x^{11} - 11313 x^{10} + 116072 x^{9} + 271530 x^{8} + 31032 x^{7} - 795618 x^{6} - 1245204 x^{5} - 20277 x^{4} + 1846080 x^{3} + 1754109 x^{2} + 374706 x - 83079 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[14, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(8210596569826381664214775617297=3^{24}\cdot 73^{6}\cdot 577^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $52.18$
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:  $3, 73, 577$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{9} + \frac{1}{3} a^{6} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{10} + \frac{1}{3} a^{7} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{11} + \frac{1}{3} a^{8} - \frac{1}{3} a^{5}$, $\frac{1}{9} a^{15} + \frac{1}{9} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{9} a^{9} - \frac{4}{9} a^{6} + \frac{1}{3} a^{3}$, $\frac{1}{9} a^{16} + \frac{1}{9} a^{13} - \frac{1}{3} a^{11} + \frac{1}{9} a^{10} - \frac{1}{3} a^{9} - \frac{4}{9} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3}$, $\frac{1}{29601350379376275358210636296466708599829529631} a^{17} - \frac{405552446902382972103662723994552162228470179}{29601350379376275358210636296466708599829529631} a^{16} - \frac{560359778145232497373378618587856414700822662}{29601350379376275358210636296466708599829529631} a^{15} - \frac{2513309748077057123603543792170242549994139231}{29601350379376275358210636296466708599829529631} a^{14} + \frac{2052123537014892373452437418318654437349128753}{29601350379376275358210636296466708599829529631} a^{13} + \frac{1271459670346822107391099457356312762085227781}{29601350379376275358210636296466708599829529631} a^{12} + \frac{7865302630748628655457801500078661551441489351}{29601350379376275358210636296466708599829529631} a^{11} - \frac{7774061424710914773968167830336703457189615677}{29601350379376275358210636296466708599829529631} a^{10} - \frac{8044832693647919872897977768245456437182075556}{29601350379376275358210636296466708599829529631} a^{9} + \frac{11794706139373305303204133173650298519890793383}{29601350379376275358210636296466708599829529631} a^{8} + \frac{10676807526689588596962347579873390829200116786}{29601350379376275358210636296466708599829529631} a^{7} - \frac{744317637366916343533064511803732213290658672}{29601350379376275358210636296466708599829529631} a^{6} + \frac{1351304200084106982611643526045421436926459116}{9867116793125425119403545432155569533276509877} a^{5} + \frac{617460252957353555185827200438261121969403090}{3289038931041808373134515144051856511092169959} a^{4} + \frac{110525108600397224598038143019831835020820844}{9867116793125425119403545432155569533276509877} a^{3} - \frac{1146606869049607331360401419891415110712650873}{3289038931041808373134515144051856511092169959} a^{2} + \frac{618013740724328308954926198157202990491716389}{3289038931041808373134515144051856511092169959} a + \frac{921415010933007007344848319345590134199762053}{3289038931041808373134515144051856511092169959}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $15$
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:  \( 1902256131.08 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T765:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 82944
The 110 conjugacy class representatives for t18n765 are not computed
Character table for t18n765 is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\), 9.9.22384826361.1

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

Sibling fields

Degree 18 siblings: 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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ R $18$ $18$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ $18$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ $18$ $18$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }$

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$3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
73Data not computed
577Data not computed