↑

Properties

Label 18.0.68808685040...5616.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{24}\cdot 3^{32}\cdot 19^{12}$
Root discriminant $126.50$
Ramified primes $2, 3, 19$
Class number $18$ (GRH)
Class group $[18]$ (GRH)
Galois group $C_2\times He_3:C_2$ (as 18T41)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![23887872, 0, 0, 3608064, 0, 0, 272484, 0, 0, -28836, 0, 0, 882, 0, 0, -42, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 42*x^15 + 882*x^12 - 28836*x^9 + 272484*x^6 + 3608064*x^3 + 23887872)
 
gp: K = bnfinit(x^18 - 42*x^15 + 882*x^12 - 28836*x^9 + 272484*x^6 + 3608064*x^3 + 23887872, 1)
 

Normalized defining polynomial

\( x^{18} - 42 x^{15} + 882 x^{12} - 28836 x^{9} + 272484 x^{6} + 3608064 x^{3} + 23887872 \)

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:  \(-68808685040765708380760273519552495616=-\,2^{24}\cdot 3^{32}\cdot 19^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $126.50$
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:  $2, 3, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3}$, $\frac{1}{3} a^{4}$, $\frac{1}{3} a^{5}$, $\frac{1}{18} a^{6}$, $\frac{1}{18} a^{7}$, $\frac{1}{18} a^{8}$, $\frac{1}{54} a^{9}$, $\frac{1}{54} a^{10}$, $\frac{1}{54} a^{11}$, $\frac{1}{5508} a^{12} + \frac{2}{459} a^{9} - \frac{1}{306} a^{6} + \frac{1}{17} a^{3} + \frac{8}{17}$, $\frac{1}{22032} a^{13} + \frac{7}{1224} a^{10} + \frac{11}{408} a^{7} - \frac{31}{204} a^{4} + \frac{25}{68} a$, $\frac{1}{264384} a^{14} - \frac{61}{14688} a^{11} + \frac{11}{4896} a^{8} + \frac{103}{816} a^{5} + \frac{25}{816} a^{2}$, $\frac{1}{20864127744} a^{15} - \frac{17495}{204550272} a^{12} + \frac{85499}{386372736} a^{9} - \frac{4685771}{193186368} a^{6} - \frac{467069}{21465152} a^{3} + \frac{951}{335393}$, $\frac{1}{83456510976} a^{16} - \frac{17495}{818201088} a^{13} - \frac{21208655}{4636472832} a^{10} - \frac{4685771}{772745472} a^{7} + \frac{20063945}{257581824} a^{4} + \frac{84086}{335393} a$, $\frac{1}{1001478131712} a^{17} - \frac{17495}{9818413056} a^{14} - \frac{50056855}{6181963776} a^{11} + \frac{27058279}{3090981888} a^{8} - \frac{409239095}{3090981888} a^{5} + \frac{419479}{4024716} a^{2}$

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

Class group and class number

$C_{18}$, which has order $18$ (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:  \( \frac{709}{1227301632} a^{15} - \frac{3683}{204550272} a^{12} + \frac{11303}{22727808} a^{9} - \frac{163703}{11363904} a^{6} + \frac{254509}{3787968} a^{3} + \frac{3473}{19729} \) (order $4$)
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:  \( 130568118565.71344 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times He_3:C_2$ (as 18T41):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 20 conjugacy class representatives for $C_2\times He_3:C_2$
Character table for $C_2\times He_3:C_2$

Intermediate fields

\(\Q(\sqrt{-1}) \), 3.1.1083.1, 6.0.75064896.1, 9.1.129610938470796864.28

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 R R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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
$2$2.6.8.1$x^{6} + 2 x^{3} + 2$$6$$1$$8$$D_{6}$$[2]_{3}^{2}$
2.12.16.3$x^{12} - 30 x^{10} - 5 x^{8} + 19 x^{4} + 30 x^{2} + 1$$6$$2$$16$$C_6\times S_3$$[2]_{3}^{6}$
$3$3.6.10.1$x^{6} - 18$$3$$2$$10$$D_{6}$$[5/2]_{2}^{2}$
3.12.22.85$x^{12} - 9 x^{11} - 33 x^{9} - 18 x^{8} + 36 x^{7} + 36 x^{6} + 27 x^{5} - 27 x^{4} + 27 x^{3} - 27 x^{2} - 27 x + 36$$6$$2$$22$$C_6\times S_3$$[2, 5/2]_{2}^{2}$
19Data not computed