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Properties

Label 18.0.14478475290...4803.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{21}\cdot 7^{12}$
Root discriminant $13.18$
Ramified primes $3, 7$
Class number $1$
Class group Trivial
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![7, 21, 21, 36, 48, 30, 25, -6, 30, -73, 99, -138, 151, -132, 114, -80, 36, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 36*x^16 - 80*x^15 + 114*x^14 - 132*x^13 + 151*x^12 - 138*x^11 + 99*x^10 - 73*x^9 + 30*x^8 - 6*x^7 + 25*x^6 + 30*x^5 + 48*x^4 + 36*x^3 + 21*x^2 + 21*x + 7)
 
gp: K = bnfinit(x^18 - 9*x^17 + 36*x^16 - 80*x^15 + 114*x^14 - 132*x^13 + 151*x^12 - 138*x^11 + 99*x^10 - 73*x^9 + 30*x^8 - 6*x^7 + 25*x^6 + 30*x^5 + 48*x^4 + 36*x^3 + 21*x^2 + 21*x + 7, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 36 x^{16} - 80 x^{15} + 114 x^{14} - 132 x^{13} + 151 x^{12} - 138 x^{11} + 99 x^{10} - 73 x^{9} + 30 x^{8} - 6 x^{7} + 25 x^{6} + 30 x^{5} + 48 x^{4} + 36 x^{3} + 21 x^{2} + 21 x + 7 \)

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:  \(-144784752906623254803=-\,3^{21}\cdot 7^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $13.18$
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:  $3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{7} a^{9} + \frac{1}{7} a^{7} - \frac{2}{7} a^{5} - \frac{1}{7} a^{3}$, $\frac{1}{7} a^{10} + \frac{1}{7} a^{8} - \frac{2}{7} a^{6} - \frac{1}{7} a^{4}$, $\frac{1}{7} a^{11} - \frac{3}{7} a^{7} + \frac{1}{7} a^{5} + \frac{1}{7} a^{3}$, $\frac{1}{7} a^{12} - \frac{3}{7} a^{8} + \frac{1}{7} a^{6} + \frac{1}{7} a^{4}$, $\frac{1}{7} a^{13} - \frac{3}{7} a^{7} + \frac{2}{7} a^{5} - \frac{3}{7} a^{3}$, $\frac{1}{49} a^{14} - \frac{3}{49} a^{13} - \frac{3}{49} a^{12} + \frac{2}{49} a^{11} - \frac{2}{49} a^{10} - \frac{1}{49} a^{9} + \frac{18}{49} a^{8} + \frac{9}{49} a^{7} - \frac{11}{49} a^{6} - \frac{16}{49} a^{5} + \frac{10}{49} a^{4} + \frac{12}{49} a^{3} + \frac{2}{7} a - \frac{2}{7}$, $\frac{1}{49} a^{15} + \frac{2}{49} a^{13} - \frac{3}{49} a^{11} + \frac{1}{49} a^{9} - \frac{19}{49} a^{7} - \frac{1}{7} a^{6} + \frac{11}{49} a^{5} - \frac{1}{7} a^{4} + \frac{1}{49} a^{3} + \frac{2}{7} a^{2} - \frac{3}{7} a + \frac{1}{7}$, $\frac{1}{49} a^{16} - \frac{1}{49} a^{13} + \frac{3}{49} a^{12} + \frac{3}{49} a^{11} - \frac{2}{49} a^{10} + \frac{2}{49} a^{9} - \frac{13}{49} a^{8} + \frac{24}{49} a^{7} - \frac{2}{49} a^{6} + \frac{18}{49} a^{5} - \frac{12}{49} a^{4} + \frac{18}{49} a^{3} - \frac{3}{7} a^{2} - \frac{3}{7} a - \frac{3}{7}$, $\frac{1}{4739231} a^{17} - \frac{16281}{4739231} a^{16} + \frac{11127}{4739231} a^{15} - \frac{16}{115591} a^{14} - \frac{61263}{4739231} a^{13} + \frac{85390}{4739231} a^{12} + \frac{289382}{4739231} a^{11} + \frac{37192}{4739231} a^{10} + \frac{272815}{4739231} a^{9} - \frac{133810}{4739231} a^{8} + \frac{1565726}{4739231} a^{7} + \frac{1482849}{4739231} a^{6} + \frac{1214250}{4739231} a^{5} - \frac{22086}{115591} a^{4} - \frac{2354910}{4739231} a^{3} + \frac{54546}{677033} a^{2} - \frac{175684}{677033} a + \frac{200006}{677033}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( -\frac{437}{2009} a^{17} + \frac{4208}{2009} a^{16} - \frac{18185}{2009} a^{15} + \frac{1090}{49} a^{14} - \frac{71384}{2009} a^{13} + \frac{89198}{2009} a^{12} - \frac{105437}{2009} a^{11} + \frac{108802}{2009} a^{10} - \frac{89284}{2009} a^{9} + \frac{68133}{2009} a^{8} - \frac{45176}{2009} a^{7} + \frac{21389}{2009} a^{6} - \frac{19764}{2009} a^{5} + \frac{27}{49} a^{4} - \frac{2241}{287} a^{3} - \frac{104}{287} a^{2} - \frac{314}{287} a - \frac{66}{41} \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1570.26677096 \)
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

\(\Q(\sqrt{-3}) \), 3.1.1323.1 x3, \(\Q(\zeta_{7})^+\), 6.0.5250987.1, 6.0.64827.1, 6.0.107163.1 x2, 9.3.2315685267.2 x3

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

Sibling fields

Degree 6 sibling: 6.0.107163.1
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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$