↑

Properties

Label 18.2.85904418583...7648.1
Degree $18$
Signature $[2, 8]$
Discriminant $2^{16}\cdot 3^{33}\cdot 11^{9}$
Root discriminant $46.03$
Ramified primes $2, 3, 11$
Class number $1$ (GRH)
Class group Trivial (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![-62702, 307158, -872982, 1741410, -2550756, 2776752, -2238954, 1352886, -655380, 288157, -119265, 40338, -9600, 1614, -216, -18, 30, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 30*x^16 - 18*x^15 - 216*x^14 + 1614*x^13 - 9600*x^12 + 40338*x^11 - 119265*x^10 + 288157*x^9 - 655380*x^8 + 1352886*x^7 - 2238954*x^6 + 2776752*x^5 - 2550756*x^4 + 1741410*x^3 - 872982*x^2 + 307158*x - 62702)
 
gp: K = bnfinit(x^18 - 9*x^17 + 30*x^16 - 18*x^15 - 216*x^14 + 1614*x^13 - 9600*x^12 + 40338*x^11 - 119265*x^10 + 288157*x^9 - 655380*x^8 + 1352886*x^7 - 2238954*x^6 + 2776752*x^5 - 2550756*x^4 + 1741410*x^3 - 872982*x^2 + 307158*x - 62702, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 30 x^{16} - 18 x^{15} - 216 x^{14} + 1614 x^{13} - 9600 x^{12} + 40338 x^{11} - 119265 x^{10} + 288157 x^{9} - 655380 x^{8} + 1352886 x^{7} - 2238954 x^{6} + 2776752 x^{5} - 2550756 x^{4} + 1741410 x^{3} - 872982 x^{2} + 307158 x - 62702 \)

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:  \(859044185836011341817275547648=2^{16}\cdot 3^{33}\cdot 11^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $46.03$
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:  $2, 3, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9}$, $\frac{1}{4} a^{11} + \frac{1}{4} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{11} + \frac{1}{8} a^{10} + \frac{3}{8} a^{9} + \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{16} a^{13} - \frac{1}{4} a^{10} + \frac{3}{16} a^{9} - \frac{3}{8} a^{8} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} + \frac{3}{8} a^{4} - \frac{1}{4} a^{3} - \frac{3}{8} a^{2} - \frac{1}{2} a - \frac{1}{8}$, $\frac{1}{32} a^{14} - \frac{1}{32} a^{13} - \frac{1}{8} a^{11} + \frac{7}{32} a^{10} + \frac{7}{32} a^{9} - \frac{1}{16} a^{8} - \frac{3}{8} a^{7} - \frac{3}{8} a^{6} - \frac{5}{16} a^{5} - \frac{5}{16} a^{4} + \frac{7}{16} a^{3} - \frac{1}{16} a^{2} - \frac{5}{16} a + \frac{1}{16}$, $\frac{1}{64} a^{15} - \frac{1}{64} a^{13} - \frac{1}{16} a^{12} + \frac{3}{64} a^{11} + \frac{7}{32} a^{10} + \frac{5}{64} a^{9} + \frac{9}{32} a^{8} + \frac{1}{8} a^{7} - \frac{11}{32} a^{6} + \frac{3}{16} a^{5} + \frac{1}{16} a^{4} + \frac{3}{16} a^{3} - \frac{3}{16} a^{2} + \frac{3}{8} a + \frac{1}{32}$, $\frac{1}{9088} a^{16} + \frac{45}{9088} a^{15} + \frac{139}{9088} a^{14} + \frac{171}{9088} a^{13} - \frac{65}{9088} a^{12} - \frac{619}{9088} a^{11} + \frac{1055}{9088} a^{10} - \frac{1297}{9088} a^{9} - \frac{651}{4544} a^{8} + \frac{689}{4544} a^{7} - \frac{145}{4544} a^{6} - \frac{327}{1136} a^{5} + \frac{35}{1136} a^{4} - \frac{501}{1136} a^{3} - \frac{563}{2272} a^{2} + \frac{433}{4544} a - \frac{95}{4544}$, $\frac{1}{370338197036383049483542150242026240} a^{17} - \frac{56018427930035299252726421281}{185169098518191524741771075121013120} a^{16} - \frac{39640177434969469136121632153961}{5786534328693485148180346097531660} a^{15} + \frac{1235191638328365934392239762907107}{185169098518191524741771075121013120} a^{14} - \frac{4375952649597768546304970075693799}{185169098518191524741771075121013120} a^{13} + \frac{1872027976141283260227556236711817}{92584549259095762370885537560506560} a^{12} - \frac{2654585716815517343618900709055461}{92584549259095762370885537560506560} a^{11} + \frac{3945941420415941514871885699127531}{37033819703638304948354215024202624} a^{10} - \frac{33702796097562251529821581002056739}{74067639407276609896708430048405248} a^{9} - \frac{26243337971920878264887720613380097}{92584549259095762370885537560506560} a^{8} + \frac{5391400873692163731773845486084917}{11573068657386970296360692195063320} a^{7} - \frac{90520194328789649456068984698260933}{185169098518191524741771075121013120} a^{6} - \frac{2073971029820986519002142109894729}{5786534328693485148180346097531660} a^{5} - \frac{1610940439114596600211108312646937}{4629227462954788118544276878025328} a^{4} - \frac{25428884597290611868432464262107029}{92584549259095762370885537560506560} a^{3} - \frac{84390544982400052945963649109060821}{185169098518191524741771075121013120} a^{2} + \frac{37629766972995032889687650204612401}{92584549259095762370885537560506560} a - \frac{78417445607301746321070360646926247}{185169098518191524741771075121013120}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 283893012.1052504 \) (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

\(\Q(\sqrt{33}) \), 3.1.108.1, 6.2.46574352.1, 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{3}$ $18$ R $18$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ $18$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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$2.9.8.1$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
$11$11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.12.6.1$x^{12} + 242 x^{8} + 21296 x^{6} + 14641 x^{4} + 1932612 x^{2} + 113379904$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$