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Properties

Label 18.0.12822537788...6875.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{43}\cdot 5^{8}$
Root discriminant $28.21$
Ramified primes $3, 5$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3^2:S_3$ (as 18T24)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![256, -2592, 11772, -31188, 52641, -59265, 47481, -31284, 20736, -14848, 10674, -6246, 3030, -1260, 468, -156, 45, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 45*x^16 - 156*x^15 + 468*x^14 - 1260*x^13 + 3030*x^12 - 6246*x^11 + 10674*x^10 - 14848*x^9 + 20736*x^8 - 31284*x^7 + 47481*x^6 - 59265*x^5 + 52641*x^4 - 31188*x^3 + 11772*x^2 - 2592*x + 256)
 
gp: K = bnfinit(x^18 - 9*x^17 + 45*x^16 - 156*x^15 + 468*x^14 - 1260*x^13 + 3030*x^12 - 6246*x^11 + 10674*x^10 - 14848*x^9 + 20736*x^8 - 31284*x^7 + 47481*x^6 - 59265*x^5 + 52641*x^4 - 31188*x^3 + 11772*x^2 - 2592*x + 256, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 45 x^{16} - 156 x^{15} + 468 x^{14} - 1260 x^{13} + 3030 x^{12} - 6246 x^{11} + 10674 x^{10} - 14848 x^{9} + 20736 x^{8} - 31284 x^{7} + 47481 x^{6} - 59265 x^{5} + 52641 x^{4} - 31188 x^{3} + 11772 x^{2} - 2592 x + 256 \)

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:  \(-128225377888491045948046875=-\,3^{43}\cdot 5^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.21$
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:  $3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{3}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{8} a^{9} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{3}{8} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{10} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} + \frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{11} - \frac{1}{4} a^{7} + \frac{1}{8} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{2}$, $\frac{1}{160} a^{12} - \frac{3}{80} a^{11} + \frac{1}{20} a^{10} - \frac{1}{32} a^{9} + \frac{1}{40} a^{7} - \frac{37}{160} a^{6} - \frac{7}{80} a^{5} - \frac{1}{5} a^{4} - \frac{31}{160} a^{3} - \frac{13}{40} a^{2} + \frac{1}{40} a + \frac{1}{5}$, $\frac{1}{160} a^{13} - \frac{1}{20} a^{11} + \frac{3}{160} a^{10} - \frac{1}{16} a^{9} + \frac{1}{40} a^{8} + \frac{27}{160} a^{7} - \frac{9}{40} a^{6} + \frac{3}{20} a^{5} + \frac{17}{160} a^{4} - \frac{29}{80} a^{3} + \frac{3}{40} a^{2} + \frac{7}{20} a + \frac{1}{5}$, $\frac{1}{160} a^{14} - \frac{1}{32} a^{11} - \frac{3}{80} a^{10} + \frac{1}{40} a^{9} + \frac{7}{160} a^{8} + \frac{9}{40} a^{7} + \frac{1}{20} a^{6} - \frac{3}{32} a^{5} - \frac{7}{80} a^{4} + \frac{1}{40} a^{3} - \frac{3}{8} a^{2} - \frac{7}{20} a - \frac{2}{5}$, $\frac{1}{160} a^{15} + \frac{1}{40} a^{11} + \frac{1}{40} a^{10} + \frac{1}{80} a^{9} - \frac{1}{40} a^{8} + \frac{7}{40} a^{7} - \frac{1}{40} a^{5} - \frac{9}{40} a^{4} - \frac{7}{32} a^{3} + \frac{1}{40} a^{2} + \frac{9}{40} a$, $\frac{1}{416372480} a^{16} - \frac{1}{52046560} a^{15} - \frac{10835}{5204656} a^{14} + \frac{215771}{104093120} a^{13} - \frac{12059}{26023280} a^{12} + \frac{1995061}{52046560} a^{11} + \frac{12875199}{208186240} a^{10} + \frac{321233}{5204656} a^{9} - \frac{4531061}{104093120} a^{8} - \frac{13788317}{104093120} a^{7} - \frac{123877}{26023280} a^{6} + \frac{1098137}{13011640} a^{5} - \frac{63610663}{416372480} a^{4} + \frac{2913603}{10409312} a^{3} + \frac{17156947}{104093120} a^{2} + \frac{3161517}{13011640} a + \frac{467617}{1626455}$, $\frac{1}{416372480} a^{17} - \frac{54179}{26023280} a^{15} - \frac{43333}{20818624} a^{14} - \frac{136907}{52046560} a^{13} - \frac{149629}{52046560} a^{12} + \frac{11658951}{208186240} a^{11} + \frac{11829}{52046560} a^{10} + \frac{4863}{20818624} a^{9} + \frac{3310919}{104093120} a^{8} + \frac{2386843}{10409312} a^{7} + \frac{5434041}{26023280} a^{6} + \frac{2906165}{83274496} a^{5} - \frac{1155789}{10409312} a^{4} - \frac{43817877}{104093120} a^{3} + \frac{475713}{13011640} a^{2} - \frac{274358}{1626455} a - \frac{487847}{1626455}$

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:  $8$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{253363503}{10409312} a^{17} + \frac{4307179551}{20818624} a^{16} - \frac{5162273991}{5204656} a^{15} + \frac{34362314355}{10409312} a^{14} - \frac{101392418595}{10409312} a^{13} + \frac{268540066197}{10409312} a^{12} - \frac{633416306799}{10409312} a^{11} + \frac{158223383043}{1301164} a^{10} - \frac{2071477875475}{10409312} a^{9} + \frac{2726143086615}{10409312} a^{8} - \frac{3890581079325}{10409312} a^{7} + \frac{5980817237307}{10409312} a^{6} - \frac{4519684708275}{5204656} a^{5} + \frac{20991269877465}{20818624} a^{4} - \frac{8089086517455}{10409312} a^{3} + \frac{1928465534649}{5204656} a^{2} - \frac{263460451443}{2602328} a + \frac{4052880846}{325291} \) (order $6$)
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:  \( 13278706.348526606 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3^2:S_3$ (as 18T24):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 10 conjugacy class representatives for $C_3^2:S_3$
Character table for $C_3^2:S_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.243.1 x3, 6.0.177147.2, 9.3.6537720751875.1

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

Sibling fields

Degree 9 siblings: data not computed
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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{9}$ R R ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
3Data not computed
$5$5.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
5.6.4.2$x^{6} - 5 x^{3} + 50$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
5.6.4.2$x^{6} - 5 x^{3} + 50$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$