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Properties

Label 18.0.10867175634...7872.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{24}\cdot 7^{12}\cdot 13^{9}$
Root discriminant $114.17$
Ramified primes $2, 3, 7, 13$
Class number $1920672$ (GRH)
Class group $[2, 2, 2, 18, 13338]$ (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![65914946488, 3006290520, 28151284764, 1059399976, 5488830594, 161406414, 650424667, 13346178, 52558377, 599534, 3060519, 10734, 130959, -168, 4077, -8, 87, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 87*x^16 - 8*x^15 + 4077*x^14 - 168*x^13 + 130959*x^12 + 10734*x^11 + 3060519*x^10 + 599534*x^9 + 52558377*x^8 + 13346178*x^7 + 650424667*x^6 + 161406414*x^5 + 5488830594*x^4 + 1059399976*x^3 + 28151284764*x^2 + 3006290520*x + 65914946488)
 
gp: K = bnfinit(x^18 + 87*x^16 - 8*x^15 + 4077*x^14 - 168*x^13 + 130959*x^12 + 10734*x^11 + 3060519*x^10 + 599534*x^9 + 52558377*x^8 + 13346178*x^7 + 650424667*x^6 + 161406414*x^5 + 5488830594*x^4 + 1059399976*x^3 + 28151284764*x^2 + 3006290520*x + 65914946488, 1)
 

Normalized defining polynomial

\( x^{18} + 87 x^{16} - 8 x^{15} + 4077 x^{14} - 168 x^{13} + 130959 x^{12} + 10734 x^{11} + 3060519 x^{10} + 599534 x^{9} + 52558377 x^{8} + 13346178 x^{7} + 650424667 x^{6} + 161406414 x^{5} + 5488830594 x^{4} + 1059399976 x^{3} + 28151284764 x^{2} + 3006290520 x + 65914946488 \)

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:  \(-10867175634578347994343560692370767872=-\,2^{18}\cdot 3^{24}\cdot 7^{12}\cdot 13^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $114.17$
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:  $2, 3, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3276=2^{2}\cdot 3^{2}\cdot 7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{3276}(1,·)$, $\chi_{3276}(2755,·)$, $\chi_{3276}(1093,·)$, $\chi_{3276}(2185,·)$, $\chi_{3276}(781,·)$, $\chi_{3276}(3067,·)$, $\chi_{3276}(1873,·)$, $\chi_{3276}(2965,·)$, $\chi_{3276}(415,·)$, $\chi_{3276}(1507,·)$, $\chi_{3276}(2599,·)$, $\chi_{3276}(625,·)$, $\chi_{3276}(883,·)$, $\chi_{3276}(1717,·)$, $\chi_{3276}(1975,·)$, $\chi_{3276}(2809,·)$, $\chi_{3276}(571,·)$, $\chi_{3276}(1663,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{12} + \frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{13} + \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{2} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{144272} a^{16} + \frac{3741}{72136} a^{15} + \frac{2165}{144272} a^{14} + \frac{32951}{72136} a^{13} + \frac{10987}{144272} a^{12} + \frac{33753}{72136} a^{11} - \frac{39775}{144272} a^{10} + \frac{4091}{36068} a^{9} - \frac{57735}{144272} a^{8} + \frac{4196}{9017} a^{7} + \frac{18419}{144272} a^{6} + \frac{2972}{9017} a^{5} - \frac{37207}{144272} a^{4} + \frac{3245}{18034} a^{3} + \frac{15039}{36068} a^{2} + \frac{278}{9017} a + \frac{47}{508}$, $\frac{1}{2427762201790120067330684931747647813746001306650764335309189968} a^{17} - \frac{724404719201999705125171156967471905811541821386430878705}{1213881100895060033665342465873823906873000653325382167654594984} a^{16} - \frac{8149252634514033752485993850930927268330745888165225273942651}{2427762201790120067330684931747647813746001306650764335309189968} a^{15} - \frac{42228230905912358434773496348948643184959334782507801445999061}{1213881100895060033665342465873823906873000653325382167654594984} a^{14} - \frac{873730911039535686635918666066006925923889246152130828908945525}{2427762201790120067330684931747647813746001306650764335309189968} a^{13} + \frac{584988799450449933757009089049954679055208966958276336295768637}{1213881100895060033665342465873823906873000653325382167654594984} a^{12} - \frac{978598776061432967095492593221562152883277056292947873583629663}{2427762201790120067330684931747647813746001306650764335309189968} a^{11} + \frac{152471134515273184284435221106003822860306789913014051771220149}{606940550447530016832671232936911953436500326662691083827297492} a^{10} + \frac{938625934396594074689977767888424377911009118016457928920335697}{2427762201790120067330684931747647813746001306650764335309189968} a^{9} - \frac{15507508429524656622852916343348310338808201246654239805093261}{151735137611882504208167808234227988359125081665672770956824373} a^{8} + \frac{1152547755258532253112233636339117897933411866076980850151123139}{2427762201790120067330684931747647813746001306650764335309189968} a^{7} - \frac{121317706518196819861450301169669163428022279884537473173748435}{303470275223765008416335616468455976718250163331345541913648746} a^{6} + \frac{1109441135860367922169065491484227929446317640042063141464018377}{2427762201790120067330684931747647813746001306650764335309189968} a^{5} - \frac{36735834804969334331271398467896120259736234422641462286965176}{151735137611882504208167808234227988359125081665672770956824373} a^{4} + \frac{43461168766950904246171938442406456817232870710216161079051419}{606940550447530016832671232936911953436500326662691083827297492} a^{3} + \frac{44228810652336364580485187888329879363921038844944142678387723}{606940550447530016832671232936911953436500326662691083827297492} a^{2} + \frac{160409458944664516079326634911159549787133914453345607902389711}{606940550447530016832671232936911953436500326662691083827297492} a - \frac{718029862849216119138096464108135770348080086510374869108301}{4274229228503732512906135443217689812933100891990782280473926}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{18}\times C_{13338}$, which has order $1920672$ (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:  \( 54408.48888868202 \) (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

\(\Q(\sqrt{-13}) \), \(\Q(\zeta_{9})^+\), 3.3.3969.1, 3.3.3969.2, \(\Q(\zeta_{7})^+\), 6.0.922529088.2, 6.0.2214992340288.7, 6.0.2214992340288.8, 6.0.337599808.1, 9.9.62523502209.1

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 R R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$2$2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
3Data not computed
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$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$$[\ ]_{3}^{3}$
$13$13.6.3.1$x^{6} - 52 x^{4} + 676 x^{2} - 79092$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
13.6.3.1$x^{6} - 52 x^{4} + 676 x^{2} - 79092$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
13.6.3.1$x^{6} - 52 x^{4} + 676 x^{2} - 79092$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$