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Properties

Label 18.0.19263351602...4928.3
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{24}\cdot 7^{15}\cdot 19^{17}$
Root discriminant $706.56$
Ramified primes $2, 3, 7, 19$
Class number $1616703552$ (GRH)
Class group $[2, 2, 12, 33681324]$ (GRH)
Galois group $C_{18}$ (as 18T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![32131593339013, 0, 179088217142121, 0, 59920722735636, 0, 4744178127122, 0, 171403421490, 0, 3369081387, 0, 38032414, 0, 242991, 0, 798, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 798*x^16 + 242991*x^14 + 38032414*x^12 + 3369081387*x^10 + 171403421490*x^8 + 4744178127122*x^6 + 59920722735636*x^4 + 179088217142121*x^2 + 32131593339013)
 
gp: K = bnfinit(x^18 + 798*x^16 + 242991*x^14 + 38032414*x^12 + 3369081387*x^10 + 171403421490*x^8 + 4744178127122*x^6 + 59920722735636*x^4 + 179088217142121*x^2 + 32131593339013, 1)
 

Normalized defining polynomial

\( x^{18} + 798 x^{16} + 242991 x^{14} + 38032414 x^{12} + 3369081387 x^{10} + 171403421490 x^{8} + 4744178127122 x^{6} + 59920722735636 x^{4} + 179088217142121 x^{2} + 32131593339013 \)

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:  \(-1926335160286020785768280733167128818936151392124928=-\,2^{18}\cdot 3^{24}\cdot 7^{15}\cdot 19^{17}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $706.56$
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:  $2, 3, 7, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4788=2^{2}\cdot 3^{2}\cdot 7\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{4788}(1,·)$, $\chi_{4788}(3055,·)$, $\chi_{4788}(4489,·)$, $\chi_{4788}(1039,·)$, $\chi_{4788}(3217,·)$, $\chi_{4788}(2803,·)$, $\chi_{4788}(559,·)$, $\chi_{4788}(2959,·)$, $\chi_{4788}(1453,·)$, $\chi_{4788}(1447,·)$, $\chi_{4788}(1063,·)$, $\chi_{4788}(2221,·)$, $\chi_{4788}(4591,·)$, $\chi_{4788}(1201,·)$, $\chi_{4788}(1261,·)$, $\chi_{4788}(439,·)$, $\chi_{4788}(505,·)$, $\chi_{4788}(1213,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7} a^{6}$, $\frac{1}{7} a^{7}$, $\frac{1}{49} a^{8} - \frac{1}{7} a^{2}$, $\frac{1}{49} a^{9} - \frac{1}{7} a^{3}$, $\frac{1}{49} a^{10} - \frac{1}{7} a^{4}$, $\frac{1}{49} a^{11} - \frac{1}{7} a^{5}$, $\frac{1}{49} a^{12}$, $\frac{1}{70217} a^{13} - \frac{107}{70217} a^{11} - \frac{257}{70217} a^{9} + \frac{454}{10031} a^{7} + \frac{2963}{10031} a^{5} - \frac{3341}{10031} a^{3} + \frac{669}{1433} a$, $\frac{1}{15237089} a^{14} - \frac{10251}{2176727} a^{12} + \frac{14498}{2176727} a^{10} + \frac{17650}{2176727} a^{8} - \frac{15135}{310961} a^{6} - \frac{32208}{310961} a^{4} + \frac{26463}{310961} a^{2} + \frac{10}{31}$, $\frac{1}{106659623} a^{15} + \frac{72}{15237089} a^{13} - \frac{9762}{2176727} a^{11} + \frac{118865}{15237089} a^{9} - \frac{126177}{2176727} a^{7} + \frac{43254}{310961} a^{5} - \frac{523508}{2176727} a^{3} + \frac{11325}{44423} a$, $\frac{1}{105479394016381591553811362016058928345309} a^{16} - \frac{149783453609446110698079597474151}{15068484859483084507687337430865561192187} a^{14} - \frac{17382808986437443052396570046958707948}{2152640694211869215383905347266508741741} a^{12} + \frac{131501526305399987355631846101880335862}{15068484859483084507687337430865561192187} a^{10} + \frac{6502060092119586371949806728656996167}{2152640694211869215383905347266508741741} a^{8} + \frac{7679910957949550178246004714083578999}{307520099173124173626272192466644105963} a^{6} - \frac{697928241264464814096563068921487225765}{2152640694211869215383905347266508741741} a^{4} - \frac{2493418779595896760448390160213754206}{6275920391288248441352493723809063387} a^{2} + \frac{1357563543318792239921941221406605}{4379567614297451808340888851227539}$, $\frac{1}{738355758114671140876679534112412498417163} a^{17} - \frac{149783453609446110698079597474151}{105479394016381591553811362016058928345309} a^{15} - \frac{30962098590938987749968418395198430}{15068484859483084507687337430865561192187} a^{13} + \frac{665852570925832082491303694840152369252}{105479394016381591553811362016058928345309} a^{11} + \frac{72016012034395167972921163054169752068}{15068484859483084507687337430865561192187} a^{9} - \frac{110104180460966118755273860050829854867}{2152640694211869215383905347266508741741} a^{7} - \frac{3061948766380400541010041416592493137341}{15068484859483084507687337430865561192187} a^{5} + \frac{16351950957590753185444589921129094099}{307520099173124173626272192466644105963} a^{3} + \frac{2285005923463150668709387600716230011}{6275920391288248441352493723809063387} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{12}\times C_{33681324}$, which has order $1616703552$ (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:  \( 1305382696.1857498 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{18}$ (as 18T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 18
The 18 conjugacy class representatives for $C_{18}$
Character table for $C_{18}$

Intermediate fields

\(\Q(\sqrt{-133}) \), 3.3.361.1, 6.0.54355325248.1, 9.9.1061871841310654257569.2

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 R R $18$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ $18$ R $18$ $18$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ $18$ $18$

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
$7$7.6.5.2$x^{6} - 7$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.2$x^{6} - 7$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.2$x^{6} - 7$$6$$1$$5$$C_6$$[\ ]_{6}$
19Data not computed