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Properties

Label 18.18.9251031023...0000.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{12}\cdot 5^{12}\cdot 41^{13}$
Root discriminant $67.84$
Ramified primes $2, 5, 41$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_3^2 : C_4$ (as 18T10)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5186, 66448, 79922, -383111, -562290, 563303, 1034639, -108479, -572263, -34860, 148967, 14111, -20806, -1797, 1595, 99, -63, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 - 63*x^16 + 99*x^15 + 1595*x^14 - 1797*x^13 - 20806*x^12 + 14111*x^11 + 148967*x^10 - 34860*x^9 - 572263*x^8 - 108479*x^7 + 1034639*x^6 + 563303*x^5 - 562290*x^4 - 383111*x^3 + 79922*x^2 + 66448*x + 5186)
 
gp: K = bnfinit(x^18 - 2*x^17 - 63*x^16 + 99*x^15 + 1595*x^14 - 1797*x^13 - 20806*x^12 + 14111*x^11 + 148967*x^10 - 34860*x^9 - 572263*x^8 - 108479*x^7 + 1034639*x^6 + 563303*x^5 - 562290*x^4 - 383111*x^3 + 79922*x^2 + 66448*x + 5186, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{17} - 63 x^{16} + 99 x^{15} + 1595 x^{14} - 1797 x^{13} - 20806 x^{12} + 14111 x^{11} + 148967 x^{10} - 34860 x^{9} - 572263 x^{8} - 108479 x^{7} + 1034639 x^{6} + 563303 x^{5} - 562290 x^{4} - 383111 x^{3} + 79922 x^{2} + 66448 x + 5186 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(925103102315013629321000000000000=2^{12}\cdot 5^{12}\cdot 41^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $67.84$
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:  $2, 5, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{5} a^{9} - \frac{1}{5} a^{8} - \frac{2}{5} a^{7} - \frac{2}{5} a^{5} + \frac{2}{5} a^{4} + \frac{2}{5} a^{2} + \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{10} + \frac{2}{5} a^{8} - \frac{2}{5} a^{7} - \frac{2}{5} a^{6} + \frac{2}{5} a^{4} + \frac{2}{5} a^{3} - \frac{2}{5} a^{2} - \frac{1}{5}$, $\frac{1}{5} a^{11} + \frac{2}{5} a^{7} + \frac{1}{5} a^{5} - \frac{2}{5} a^{4} - \frac{2}{5} a^{3} + \frac{1}{5} a^{2} + \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{10} a^{12} - \frac{1}{10} a^{9} - \frac{1}{5} a^{8} + \frac{1}{5} a^{7} + \frac{1}{10} a^{6} - \frac{2}{5} a^{4} + \frac{1}{10} a^{3} - \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{20} a^{13} - \frac{1}{20} a^{12} - \frac{1}{10} a^{11} + \frac{1}{20} a^{10} + \frac{1}{20} a^{9} - \frac{1}{5} a^{8} + \frac{7}{20} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{7}{20} a^{4} + \frac{7}{20} a^{3} + \frac{1}{5} a^{2} - \frac{1}{10} a + \frac{3}{10}$, $\frac{1}{2460} a^{14} + \frac{23}{2460} a^{13} + \frac{22}{615} a^{12} + \frac{27}{820} a^{11} + \frac{221}{2460} a^{10} - \frac{37}{1230} a^{9} - \frac{39}{164} a^{8} - \frac{49}{164} a^{7} + \frac{298}{615} a^{6} + \frac{9}{164} a^{5} - \frac{241}{492} a^{4} - \frac{83}{410} a^{3} - \frac{27}{82} a^{2} + \frac{63}{410} a + \frac{62}{615}$, $\frac{1}{4920} a^{15} - \frac{1}{4920} a^{14} - \frac{19}{984} a^{13} + \frac{1}{82} a^{12} - \frac{1}{4920} a^{11} - \frac{89}{4920} a^{10} + \frac{7}{410} a^{9} - \frac{97}{328} a^{8} + \frac{2227}{4920} a^{7} - \frac{169}{410} a^{6} - \frac{263}{4920} a^{5} + \frac{331}{1640} a^{4} - \frac{69}{328} a^{3} - \frac{223}{820} a^{2} - \frac{599}{2460} a + \frac{33}{820}$, $\frac{1}{4920} a^{16} - \frac{1}{120} a^{13} - \frac{103}{4920} a^{12} + \frac{51}{820} a^{11} + \frac{301}{4920} a^{10} + \frac{9}{328} a^{9} + \frac{421}{1230} a^{8} - \frac{251}{4920} a^{7} - \frac{281}{4920} a^{6} - \frac{289}{2460} a^{5} + \frac{27}{82} a^{4} - \frac{573}{1640} a^{3} + \frac{49}{615} a^{2} + \frac{89}{1230} a - \frac{17}{820}$, $\frac{1}{289548387845959100796468240} a^{17} + \frac{659087230233451085575}{19303225856397273386431216} a^{16} + \frac{831056776227677509231}{24129032320496591733039020} a^{15} + \frac{3618007395843396497023}{96516129281986366932156080} a^{14} + \frac{354180960871594941503073}{24129032320496591733039020} a^{13} - \frac{6978159192177653682034819}{289548387845959100796468240} a^{12} + \frac{1071366000994737110292553}{289548387845959100796468240} a^{11} - \frac{3445733697972563640327443}{72387096961489775199117060} a^{10} + \frac{6597957118505994466627559}{96516129281986366932156080} a^{9} + \frac{107089130449727432903528923}{289548387845959100796468240} a^{8} - \frac{2603444795322212626338452}{18096774240372443799779265} a^{7} + \frac{487828377381310063873765}{19303225856397273386431216} a^{6} - \frac{5512478409011680055196383}{12064516160248295866519510} a^{5} - \frac{189422875912961548650655}{1412431160224190735592528} a^{4} - \frac{23253993173366716530193697}{57909677569191820159293648} a^{3} - \frac{1219508905107324733442458}{18096774240372443799779265} a^{2} + \frac{21688178997513566954043137}{48258064640993183466078040} a + \frac{2141468357499486372100795}{28954838784595910079646824}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  $17$
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:  \( 156794220435 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3:S_3.C_2$ (as 18T10):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 6 conjugacy class representatives for $C_3^2 : C_4$
Character table for $C_3^2 : C_4$

Intermediate fields

\(\Q(\sqrt{41}) \), 6.6.7064402500.1 x2, 6.6.1130304400.1 x2, 9.9.4750104241000000.1

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

Sibling fields

Galois closure: data not computed
Degree 6 siblings: data not computed
Degree 9 sibling: data not computed
Degree 12 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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$5$5.3.2.1$x^{3} - 5$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
5.3.2.1$x^{3} - 5$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$41$41.2.1.1$x^{2} - 41$$2$$1$$1$$C_2$$[\ ]_{2}$
41.4.3.1$x^{4} - 41$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.3.1$x^{4} - 41$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.3.1$x^{4} - 41$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.3.1$x^{4} - 41$$4$$1$$3$$C_4$$[\ ]_{4}$