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Properties

Label 18.0.11990701314...5739.3
Degree $18$
Signature $[0, 9]$
Discriminant $-\,7^{12}\cdot 59^{9}$
Root discriminant $28.11$
Ramified primes $7, 59$
Class number $9$
Class group $[9]$
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![5249, -25103, 64007, -111112, 136638, -117510, 69340, -27325, 7035, -2321, 2121, -1472, 569, -30, -41, 13, 9, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 + 9*x^16 + 13*x^15 - 41*x^14 - 30*x^13 + 569*x^12 - 1472*x^11 + 2121*x^10 - 2321*x^9 + 7035*x^8 - 27325*x^7 + 69340*x^6 - 117510*x^5 + 136638*x^4 - 111112*x^3 + 64007*x^2 - 25103*x + 5249)
 
gp: K = bnfinit(x^18 - 4*x^17 + 9*x^16 + 13*x^15 - 41*x^14 - 30*x^13 + 569*x^12 - 1472*x^11 + 2121*x^10 - 2321*x^9 + 7035*x^8 - 27325*x^7 + 69340*x^6 - 117510*x^5 + 136638*x^4 - 111112*x^3 + 64007*x^2 - 25103*x + 5249, 1)
 

Normalized defining polynomial

\( x^{18} - 4 x^{17} + 9 x^{16} + 13 x^{15} - 41 x^{14} - 30 x^{13} + 569 x^{12} - 1472 x^{11} + 2121 x^{10} - 2321 x^{9} + 7035 x^{8} - 27325 x^{7} + 69340 x^{6} - 117510 x^{5} + 136638 x^{4} - 111112 x^{3} + 64007 x^{2} - 25103 x + 5249 \)

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:  \(-119907013147065124216135739=-\,7^{12}\cdot 59^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.11$
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:  $7, 59$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{58} a^{13} + \frac{9}{58} a^{12} - \frac{6}{29} a^{11} + \frac{11}{58} a^{10} + \frac{21}{58} a^{9} + \frac{25}{58} a^{8} + \frac{17}{58} a^{7} - \frac{6}{29} a^{6} + \frac{11}{58} a^{5} - \frac{13}{29} a^{4} - \frac{9}{58} a^{3} + \frac{27}{58} a^{2} - \frac{12}{29} a - \frac{1}{2}$, $\frac{1}{116} a^{14} + \frac{23}{116} a^{12} - \frac{55}{116} a^{11} - \frac{5}{29} a^{10} - \frac{12}{29} a^{9} - \frac{17}{58} a^{8} - \frac{49}{116} a^{7} - \frac{55}{116} a^{6} + \frac{49}{116} a^{5} + \frac{51}{116} a^{4} + \frac{25}{58} a^{3} - \frac{35}{116} a^{2} + \frac{13}{116} a + \frac{1}{4}$, $\frac{1}{3480} a^{15} + \frac{1}{1160} a^{14} + \frac{23}{3480} a^{13} - \frac{103}{348} a^{12} - \frac{881}{3480} a^{11} + \frac{107}{290} a^{10} + \frac{607}{1740} a^{9} - \frac{499}{3480} a^{8} - \frac{227}{580} a^{7} - \frac{3}{10} a^{6} + \frac{389}{1740} a^{5} + \frac{1}{8} a^{4} + \frac{77}{1160} a^{3} + \frac{61}{435} a^{2} + \frac{139}{435} a - \frac{53}{120}$, $\frac{1}{90480} a^{16} - \frac{1}{7540} a^{15} - \frac{161}{45240} a^{14} - \frac{59}{18096} a^{13} + \frac{3469}{90480} a^{12} + \frac{2533}{30160} a^{11} - \frac{5303}{45240} a^{10} - \frac{9469}{90480} a^{9} + \frac{14441}{30160} a^{8} - \frac{4599}{15080} a^{7} + \frac{13469}{45240} a^{6} + \frac{3007}{6032} a^{5} - \frac{1319}{15080} a^{4} + \frac{21023}{90480} a^{3} + \frac{6623}{22620} a^{2} + \frac{35483}{90480} a - \frac{55}{208}$, $\frac{1}{16174745026409120296011360} a^{17} + \frac{1094147244644296411}{16174745026409120296011360} a^{16} + \frac{311075461528819497239}{2695790837734853382668560} a^{15} + \frac{9891266464139730655103}{3234949005281824059202272} a^{14} - \frac{151991613668187001201}{27887491424843310855192} a^{13} - \frac{379522872797068752410689}{2695790837734853382668560} a^{12} + \frac{177202138521151562195981}{414737051959208212718240} a^{11} - \frac{5976143946827084146327}{42903832961297401315680} a^{10} + \frac{187826586220966723538207}{505460782075285009250355} a^{9} + \frac{2437843030634320389504947}{16174745026409120296011360} a^{8} - \frac{854302536428737545753407}{4043686256602280074002840} a^{7} + \frac{14944477633690246471987}{1244211155877624638154720} a^{6} - \frac{3818078223607529519187683}{16174745026409120296011360} a^{5} - \frac{7148763305804287969838839}{16174745026409120296011360} a^{4} - \frac{1797186541831279488086987}{16174745026409120296011360} a^{3} + \frac{310102636776226914246673}{1078316335093941353067424} a^{2} + \frac{144594584494793462201651}{2021843128301140037001420} a + \frac{51469395109883167309969}{557749828496866217103840}$

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

Class group and class number

$C_{9}$, which has order $9$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 48431.5389411 \)
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

\(\Q(\sqrt{-59}) \), 3.1.2891.3 x3, \(\Q(\zeta_{7})^+\), 6.0.493114979.2, 6.0.10063571.2 x2, 6.0.493114979.4, 9.3.24162633971.3 x3

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

Sibling fields

Degree 6 sibling: 6.0.10063571.2
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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ R

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
$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}$
$59$59.6.3.2$x^{6} - 3481 x^{2} + 3491443$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
59.6.3.2$x^{6} - 3481 x^{2} + 3491443$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
59.6.3.2$x^{6} - 3481 x^{2} + 3491443$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$