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Properties

Label 18.0.19263351602...4928.2
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 $1465818048$ (GRH)
Class group $[2, 2, 12, 30537876]$ (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![18624704126077, 0, 39871239158709, 0, 22440538637328, 0, 2538023959562, 0, 119120371902, 0, 2831149587, 0, 36093274, 0, 242991, 0, 798, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 798*x^16 + 242991*x^14 + 36093274*x^12 + 2831149587*x^10 + 119120371902*x^8 + 2538023959562*x^6 + 22440538637328*x^4 + 39871239158709*x^2 + 18624704126077)
 
gp: K = bnfinit(x^18 + 798*x^16 + 242991*x^14 + 36093274*x^12 + 2831149587*x^10 + 119120371902*x^8 + 2538023959562*x^6 + 22440538637328*x^4 + 39871239158709*x^2 + 18624704126077, 1)
 

Normalized defining polynomial

\( x^{18} + 798 x^{16} + 242991 x^{14} + 36093274 x^{12} + 2831149587 x^{10} + 119120371902 x^{8} + 2538023959562 x^{6} + 22440538637328 x^{4} + 39871239158709 x^{2} + 18624704126077 \)

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$$\chi_{4788}(1,·)$, $\chi_{4788}(2635,·)$, $\chi_{4788}(2893,·)$, $\chi_{4788}(1261,·)$, $\chi_{4788}(1363,·)$, $\chi_{4788}(4591,·)$, $\chi_{4788}(25,·)$, $\chi_{4788}(4399,·)$, $\chi_{4788}(4639,·)$, $\chi_{4788}(1063,·)$, $\chi_{4788}(3049,·)$, $\chi_{4788}(4651,·)$, $\chi_{4788}(2797,·)$, $\chi_{4788}(3631,·)$, $\chi_{4788}(625,·)$, $\chi_{4788}(4405,·)$, $\chi_{4788}(505,·)$, $\chi_{4788}(559,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7} a^{6}$, $\frac{1}{7} a^{7}$, $\frac{1}{49} a^{8} - \frac{2}{7} a^{2}$, $\frac{1}{49} a^{9} - \frac{2}{7} a^{3}$, $\frac{1}{49} a^{10} - \frac{2}{7} a^{4}$, $\frac{1}{49} a^{11} - \frac{2}{7} a^{5}$, $\frac{1}{49} a^{12}$, $\frac{1}{53459} a^{13} - \frac{1}{53459} a^{11} + \frac{452}{53459} a^{9} - \frac{349}{7637} a^{7} + \frac{3383}{7637} a^{5} + \frac{1462}{7637} a^{3} + \frac{124}{1091} a$, $\frac{1}{374213} a^{14} - \frac{156}{53459} a^{12} - \frac{403}{53459} a^{10} - \frac{349}{53459} a^{8} - \frac{296}{7637} a^{6} + \frac{3326}{7637} a^{4} - \frac{3149}{7637} a^{2}$, $\frac{1}{2619491} a^{15} + \frac{2}{374213} a^{13} - \frac{236}{53459} a^{11} + \frac{2334}{374213} a^{9} - \frac{3070}{53459} a^{7} + \frac{2958}{7637} a^{5} - \frac{24174}{53459} a^{3} + \frac{461}{1091} a$, $\frac{1}{15008775345598724326652728179380524624749407} a^{16} + \frac{2332210345541489206789494876558706100}{2144110763656960618093246882768646374964201} a^{14} + \frac{643763842319735249071899167712117961148}{306301537665280088299035268966949482137743} a^{12} + \frac{3414747621453336895204572309828028909125}{2144110763656960618093246882768646374964201} a^{10} + \frac{2896705768648139872726587449748247105396}{306301537665280088299035268966949482137743} a^{8} - \frac{284919036529505083996383409952371207393}{43757362523611441185576466995278497448249} a^{6} - \frac{975314537574196583550783623458381905434}{2973801336556117362126556009387859049881} a^{4} + \frac{167146003725475155532003927579073416194}{893007398441049820113805448883234641801} a^{2} - \frac{276953184623673752103662918496104222}{818521905078872429068565947647327811}$, $\frac{1}{105061427419191070286569097255663672373245849} a^{17} + \frac{2332210345541489206789494876558706100}{15008775345598724326652728179380524624749407} a^{15} - \frac{15146291268757056328296420143980926707}{2144110763656960618093246882768646374964201} a^{13} - \frac{123244969014261540523723459561014771220637}{15008775345598724326652728179380524624749407} a^{11} + \frac{11370863051929706130873450705741031932679}{2144110763656960618093246882768646374964201} a^{9} + \frac{17138956756884452313586179917616295905364}{306301537665280088299035268966949482137743} a^{7} - \frac{1228031189063645341966409504051902990071}{20816609355892821534885892065715013349167} a^{5} + \frac{1075073485089176886952073843126423887282}{43757362523611441185576466995278497448249} a^{3} - \frac{307313843999657268687283817116366175893}{893007398441049820113805448883234641801} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{12}\times C_{30537876}$, which has order $1465818048$ (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:  \( 824118633.8968437 \) (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, 9.9.1061871841310654257569.5

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 $18$ $18$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ $18$ $18$

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$7.6.5.3$x^{6} - 112$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.3$x^{6} - 112$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.3$x^{6} - 112$$6$$1$$5$$C_6$$[\ ]_{6}$
19Data not computed