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Properties

Label 18.0.32680784318...8587.3
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{39}\cdot 7^{12}\cdot 17^{12}$
Root discriminant $261.49$
Ramified primes $3, 7, 17$
Class number $15309$ (GRH)
Class group $[3, 27, 189]$ (GRH)
Galois group $C_3\times C_3:S_3$ (as 18T23)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![174676879, 0, 0, -127849638, 0, 0, 44713203, 0, 0, 467768, 0, 0, -7815, 0, 0, -21, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 21*x^15 - 7815*x^12 + 467768*x^9 + 44713203*x^6 - 127849638*x^3 + 174676879)
 
gp: K = bnfinit(x^18 - 21*x^15 - 7815*x^12 + 467768*x^9 + 44713203*x^6 - 127849638*x^3 + 174676879, 1)
 

Normalized defining polynomial

\( x^{18} - 21 x^{15} - 7815 x^{12} + 467768 x^{9} + 44713203 x^{6} - 127849638 x^{3} + 174676879 \)

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:  \(-32680784318068732897853908300449068156188587=-\,3^{39}\cdot 7^{12}\cdot 17^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $261.49$
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:  $3, 7, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{3} a^{5} + \frac{1}{3} a^{2}$, $\frac{1}{9} a^{6} - \frac{1}{9} a^{3} - \frac{2}{9}$, $\frac{1}{9} a^{7} - \frac{1}{9} a^{4} - \frac{2}{9} a$, $\frac{1}{27} a^{8} - \frac{1}{27} a^{7} + \frac{1}{27} a^{6} + \frac{2}{27} a^{5} - \frac{2}{27} a^{4} + \frac{2}{27} a^{3} + \frac{1}{27} a^{2} - \frac{1}{27} a + \frac{1}{27}$, $\frac{1}{513} a^{9} + \frac{1}{19} a^{6} - \frac{10}{171} a^{3} - \frac{164}{513}$, $\frac{1}{513} a^{10} + \frac{1}{19} a^{7} - \frac{10}{171} a^{4} - \frac{164}{513} a$, $\frac{1}{1539} a^{11} - \frac{1}{1539} a^{10} + \frac{1}{1539} a^{9} + \frac{1}{57} a^{8} - \frac{1}{57} a^{7} + \frac{1}{57} a^{6} - \frac{10}{513} a^{5} + \frac{10}{513} a^{4} - \frac{10}{513} a^{3} + \frac{349}{1539} a^{2} - \frac{349}{1539} a + \frac{349}{1539}$, $\frac{1}{56943} a^{12} - \frac{29}{56943} a^{9} + \frac{341}{18981} a^{6} + \frac{9211}{56943} a^{3} + \frac{18931}{56943}$, $\frac{1}{170829} a^{13} + \frac{1}{170829} a^{12} - \frac{140}{170829} a^{10} - \frac{140}{170829} a^{9} + \frac{1451}{56943} a^{7} + \frac{1451}{56943} a^{6} + \frac{6214}{170829} a^{4} + \frac{6214}{170829} a^{3} + \frac{81424}{170829} a + \frac{81424}{170829}$, $\frac{1}{170829} a^{14} - \frac{1}{170829} a^{12} - \frac{29}{170829} a^{11} - \frac{1}{1539} a^{10} - \frac{82}{170829} a^{9} + \frac{341}{56943} a^{8} + \frac{10}{513} a^{7} + \frac{769}{56943} a^{6} - \frac{9770}{170829} a^{5} + \frac{16}{171} a^{4} + \frac{25754}{170829} a^{3} - \frac{56993}{170829} a^{2} - \frac{292}{1539} a + \frac{24581}{170829}$, $\frac{1}{2653729263351} a^{15} + \frac{12718457}{2653729263351} a^{12} + \frac{1467830425}{2653729263351} a^{9} - \frac{49817893805}{2653729263351} a^{6} + \frac{366070091210}{2653729263351} a^{3} + \frac{1164695236042}{2653729263351}$, $\frac{1}{4450303974639627} a^{16} + \frac{1}{7961187790053} a^{15} - \frac{3342716047}{4450303974639627} a^{13} + \frac{12718457}{7961187790053} a^{12} - \frac{4683799499}{4450303974639627} a^{10} + \frac{6640791952}{7961187790053} a^{9} + \frac{125820918937495}{4450303974639627} a^{7} + \frac{384710874463}{7961187790053} a^{6} - \frac{607129474847608}{4450303974639627} a^{4} + \frac{800598859478}{7961187790053} a^{3} - \frac{1272002358832862}{4450303974639627} a + \frac{3264917616004}{7961187790053}$, $\frac{1}{2487719921823551493} a^{17} - \frac{1}{7961187790053} a^{15} + \frac{130869846683}{67235673562798689} a^{14} - \frac{12718457}{7961187790053} a^{12} - \frac{666811727889647}{2487719921823551493} a^{11} - \frac{1}{1539} a^{10} - \frac{1467830425}{7961187790053} a^{9} + \frac{41474657021119714}{2487719921823551493} a^{8} - \frac{28}{513} a^{7} + \frac{49817893805}{7961187790053} a^{6} + \frac{143215615922381576}{2487719921823551493} a^{5} - \frac{28}{513} a^{4} - \frac{366070091210}{7961187790053} a^{3} + \frac{1074412493003110306}{2487719921823551493} a^{2} - \frac{406}{1539} a - \frac{1164695236042}{7961187790053}$

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

Class group and class number

$C_{3}\times C_{27}\times C_{189}$, which has order $15309$ (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:  \( -\frac{62}{4400877717} a^{15} + \frac{49}{162995471} a^{12} + \frac{157621}{1466959239} a^{9} - \frac{28559699}{4400877717} a^{6} - \frac{916925509}{1466959239} a^{3} + \frac{675665378}{488986413} \) (order $6$)
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:  \( 267000059795.07947 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_3:S_3$ (as 18T23):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 18 conjugacy class representatives for $C_3\times C_3:S_3$
Character table for $C_3\times C_3:S_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.70227.1 x3, 6.0.47258883.3, Deg 6, Deg 6, 6.0.14795494587.3

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/19.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{9}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
3Data not computed
$7$7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
$17$17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$