↑

Properties

Label 18.2.14401256115...8672.1
Degree $18$
Signature $[2, 8]$
Discriminant $2^{16}\cdot 3^{32}\cdot 17^{9}$
Root discriminant $53.83$
Ramified primes $2, 3, 17$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2\times D_9:C_3$ (as 18T45)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-388034, -410058, -92772, 684582, 369702, -500040, -309084, 239796, 118278, -68955, -28413, 14004, 4296, -2142, -306, 204, 0, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 204*x^15 - 306*x^14 - 2142*x^13 + 4296*x^12 + 14004*x^11 - 28413*x^10 - 68955*x^9 + 118278*x^8 + 239796*x^7 - 309084*x^6 - 500040*x^5 + 369702*x^4 + 684582*x^3 - 92772*x^2 - 410058*x - 388034)
 
gp: K = bnfinit(x^18 - 9*x^17 + 204*x^15 - 306*x^14 - 2142*x^13 + 4296*x^12 + 14004*x^11 - 28413*x^10 - 68955*x^9 + 118278*x^8 + 239796*x^7 - 309084*x^6 - 500040*x^5 + 369702*x^4 + 684582*x^3 - 92772*x^2 - 410058*x - 388034, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 204 x^{15} - 306 x^{14} - 2142 x^{13} + 4296 x^{12} + 14004 x^{11} - 28413 x^{10} - 68955 x^{9} + 118278 x^{8} + 239796 x^{7} - 309084 x^{6} - 500040 x^{5} + 369702 x^{4} + 684582 x^{3} - 92772 x^{2} - 410058 x - 388034 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(14401256115536510106932410908672=2^{16}\cdot 3^{32}\cdot 17^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $53.83$
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:  $2, 3, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{9} + \frac{1}{3} a^{7} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{6} + \frac{1}{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{7} + \frac{1}{3} a$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{8} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{9} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{16} - \frac{1}{3} a^{8} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{604067689320663923581136882037168468191899935} a^{17} - \frac{7461371039708652505624215011032996328593003}{604067689320663923581136882037168468191899935} a^{16} + \frac{1790321856200070825629276310893520832977292}{604067689320663923581136882037168468191899935} a^{15} + \frac{18832625947229450931688251198789438610392262}{201355896440221307860378960679056156063966645} a^{14} - \frac{9953209046674654143793868518622844355260347}{120813537864132784716227376407433693638379987} a^{13} - \frac{19396376629809761277739433902056343234869322}{604067689320663923581136882037168468191899935} a^{12} - \frac{64393364701738271660013507269902237240711391}{604067689320663923581136882037168468191899935} a^{11} + \frac{1147757154819740332181243347711184326539606}{201355896440221307860378960679056156063966645} a^{10} - \frac{4180610882420764914490527041103162644470726}{40271179288044261572075792135811231212793329} a^{9} - \frac{12157574678206164239306977804240373931421108}{120813537864132784716227376407433693638379987} a^{8} - \frac{49937693353621097792062359899959458416827074}{201355896440221307860378960679056156063966645} a^{7} + \frac{15538429391139195842836797476605085816608808}{201355896440221307860378960679056156063966645} a^{6} + \frac{6694742244780666061097657006499055740101392}{40271179288044261572075792135811231212793329} a^{5} - \frac{37158522519498628612073864051631779699366302}{120813537864132784716227376407433693638379987} a^{4} - \frac{210919774250118894098751886751031073622879458}{604067689320663923581136882037168468191899935} a^{3} + \frac{1061966084073575418493982577196034735728654}{604067689320663923581136882037168468191899935} a^{2} - \frac{154966671976211995483521293177862742289767643}{604067689320663923581136882037168468191899935} a + \frac{277107651428515565774300183816452045202445964}{604067689320663923581136882037168468191899935}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $9$
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:  \( 344611950.81361854 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times D_9:C_3$ (as 18T45):

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 $C_2\times D_9:C_3$
Character table for $C_2\times D_9:C_3$

Intermediate fields

\(\Q(\sqrt{17}) \), 3.1.108.1, 6.2.57305232.5, 9.1.11019960576.1

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

Sibling fields

Degree 18 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 R R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{3}$ $18$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ $18$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.9.8.1$x^{9} - 2$$9$$1$$8$$(C_9:C_3):C_2$$[\ ]_{9}^{6}$
2.9.8.1$x^{9} - 2$$9$$1$$8$$(C_9:C_3):C_2$$[\ ]_{9}^{6}$
3Data not computed
$17$17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$