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Properties

Label 18.18.6496655965...0992.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{24}\cdot 3^{17}\cdot 7^{10}\cdot 101^{6}$
Root discriminant $97.63$
Ramified primes $2, 3, 7, 101$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_3:S_3:S_4$ (as 18T153)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-17352, 58512, 287632, -222560, -1176184, -276264, 1110410, 450704, -460148, -201906, 104926, 41214, -14609, -4226, 1223, 210, -55, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 - 55*x^16 + 210*x^15 + 1223*x^14 - 4226*x^13 - 14609*x^12 + 41214*x^11 + 104926*x^10 - 201906*x^9 - 460148*x^8 + 450704*x^7 + 1110410*x^6 - 276264*x^5 - 1176184*x^4 - 222560*x^3 + 287632*x^2 + 58512*x - 17352)
 
gp: K = bnfinit(x^18 - 4*x^17 - 55*x^16 + 210*x^15 + 1223*x^14 - 4226*x^13 - 14609*x^12 + 41214*x^11 + 104926*x^10 - 201906*x^9 - 460148*x^8 + 450704*x^7 + 1110410*x^6 - 276264*x^5 - 1176184*x^4 - 222560*x^3 + 287632*x^2 + 58512*x - 17352, 1)
 

Normalized defining polynomial

\( x^{18} - 4 x^{17} - 55 x^{16} + 210 x^{15} + 1223 x^{14} - 4226 x^{13} - 14609 x^{12} + 41214 x^{11} + 104926 x^{10} - 201906 x^{9} - 460148 x^{8} + 450704 x^{7} + 1110410 x^{6} - 276264 x^{5} - 1176184 x^{4} - 222560 x^{3} + 287632 x^{2} + 58512 x - 17352 \)

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:  \(649665596506914089007541014591700992=2^{24}\cdot 3^{17}\cdot 7^{10}\cdot 101^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $97.63$
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:  $2, 3, 7, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8}$, $\frac{1}{6} a^{13} - \frac{1}{6} a^{10} - \frac{1}{2} a^{8} - \frac{1}{6} a^{7} - \frac{1}{3} a^{4} - \frac{1}{3} a$, $\frac{1}{6} a^{14} - \frac{1}{6} a^{11} + \frac{1}{3} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{2}$, $\frac{1}{12} a^{15} - \frac{1}{12} a^{13} + \frac{1}{6} a^{12} - \frac{1}{4} a^{11} - \frac{1}{6} a^{10} - \frac{1}{12} a^{9} - \frac{1}{6} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{6} a^{3} - \frac{1}{3} a$, $\frac{1}{24} a^{16} + \frac{1}{24} a^{14} + \frac{1}{8} a^{12} - \frac{1}{6} a^{11} - \frac{5}{24} a^{10} + \frac{1}{12} a^{8} - \frac{1}{2} a^{7} + \frac{1}{6} a^{5} + \frac{1}{12} a^{4} - \frac{1}{3} a^{2} + \frac{1}{6} a$, $\frac{1}{32133051598990240216980312} a^{17} - \frac{153617339703533227427}{765072657118815243261436} a^{16} - \frac{551188365690190942577303}{32133051598990240216980312} a^{15} + \frac{608079879738613429069967}{16066525799495120108490156} a^{14} + \frac{180038439043892915167909}{4590435942712891459568616} a^{13} - \frac{1793781710752103131091549}{16066525799495120108490156} a^{12} - \frac{2225007673822267549632195}{10711017199663413405660104} a^{11} + \frac{3868481664078055160434075}{16066525799495120108490156} a^{10} - \frac{2024548510820231305504943}{16066525799495120108490156} a^{9} + \frac{2600503896904180373621747}{8033262899747560054245078} a^{8} - \frac{890567715973696625661967}{8033262899747560054245078} a^{7} + \frac{1273878426571660715129630}{4016631449873780027122539} a^{6} - \frac{1916204763504049904812233}{5355508599831706702830052} a^{5} + \frac{2919807472803498309641737}{8033262899747560054245078} a^{4} - \frac{373084350835479272610262}{4016631449873780027122539} a^{3} + \frac{2706244593979963584156551}{8033262899747560054245078} a^{2} + \frac{1087213709694016660896362}{4016631449873780027122539} a - \frac{443755395594167214554125}{1338877149957926675707513}$

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:  $17$
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:  \( 11607696293900 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3:S_3:S_4$ (as 18T153):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 432
The 20 conjugacy class representatives for $C_3:S_3:S_4$
Character table for $C_3:S_3:S_4$

Intermediate fields

3.3.404.1, 6.6.3454956288.1, 9.9.5539939488857088.1

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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$2.6.8.3$x^{6} + 2 x^{3} + 6$$6$$1$$8$$D_{6}$$[2]_{3}^{2}$
2.6.8.3$x^{6} + 2 x^{3} + 6$$6$$1$$8$$D_{6}$$[2]_{3}^{2}$
2.6.8.3$x^{6} + 2 x^{3} + 6$$6$$1$$8$$D_{6}$$[2]_{3}^{2}$
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.12.14.3$x^{12} - 12 x^{11} + 3 x^{10} - 9 x^{7} - 6 x^{6} - 9 x^{3} + 9 x^{2} - 9$$6$$2$$14$$S_3 \times C_4$$[3/2]_{2}^{4}$
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.12.10.6$x^{12} - 217 x^{6} + 11907$$6$$2$$10$$C_{12}$$[\ ]_{6}^{2}$
101Data not computed