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Properties

Label 18.0.41155853096...6416.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{25}\cdot 17^{9}$
Root discriminant $30.10$
Ramified primes $2, 3, 17$
Class number $18$ (GRH)
Class group $[3, 6]$ (GRH)
Galois group $S_3^2$ (as 18T11)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![484, 792, 1608, -816, 2646, -3468, 3981, -1077, -519, 446, 1545, -909, -90, -123, 129, 6, -9, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 9*x^16 + 6*x^15 + 129*x^14 - 123*x^13 - 90*x^12 - 909*x^11 + 1545*x^10 + 446*x^9 - 519*x^8 - 1077*x^7 + 3981*x^6 - 3468*x^5 + 2646*x^4 - 816*x^3 + 1608*x^2 + 792*x + 484)
 
gp: K = bnfinit(x^18 - 3*x^17 - 9*x^16 + 6*x^15 + 129*x^14 - 123*x^13 - 90*x^12 - 909*x^11 + 1545*x^10 + 446*x^9 - 519*x^8 - 1077*x^7 + 3981*x^6 - 3468*x^5 + 2646*x^4 - 816*x^3 + 1608*x^2 + 792*x + 484, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 9 x^{16} + 6 x^{15} + 129 x^{14} - 123 x^{13} - 90 x^{12} - 909 x^{11} + 1545 x^{10} + 446 x^{9} - 519 x^{8} - 1077 x^{7} + 3981 x^{6} - 3468 x^{5} + 2646 x^{4} - 816 x^{3} + 1608 x^{2} + 792 x + 484 \)

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:  \(-411558530965263777632956416=-\,2^{12}\cdot 3^{25}\cdot 17^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.10$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{11} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{12} - \frac{1}{4} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{13816} a^{16} + \frac{1233}{13816} a^{15} + \frac{597}{6908} a^{14} + \frac{1427}{13816} a^{13} - \frac{59}{13816} a^{12} - \frac{1571}{6908} a^{11} + \frac{1015}{13816} a^{10} - \frac{1029}{13816} a^{9} + \frac{2229}{6908} a^{8} + \frac{5529}{13816} a^{7} + \frac{3733}{13816} a^{6} - \frac{1295}{6908} a^{5} - \frac{27}{6908} a^{4} + \frac{425}{1727} a^{3} - \frac{546}{1727} a^{2} - \frac{438}{1727} a - \frac{153}{314}$, $\frac{1}{554499101708376997107131980696} a^{17} - \frac{9206749683426129085310353}{554499101708376997107131980696} a^{16} - \frac{1352509982808398203592969087}{138624775427094249276782995174} a^{15} + \frac{48620567129833034643360460911}{554499101708376997107131980696} a^{14} - \frac{109083837251578149657805459405}{554499101708376997107131980696} a^{13} + \frac{30853266674267952944162099911}{138624775427094249276782995174} a^{12} + \frac{7843969562229800790803882221}{50409009246216090646102907336} a^{11} + \frac{1371930800009055286653360991}{50409009246216090646102907336} a^{10} - \frac{36037181034483953417419666543}{138624775427094249276782995174} a^{9} + \frac{229670256806785627647980060997}{554499101708376997107131980696} a^{8} - \frac{73197146475994546694302797561}{554499101708376997107131980696} a^{7} + \frac{51557780023956960727340025547}{138624775427094249276782995174} a^{6} - \frac{82734706593514750513376332287}{277249550854188498553565990348} a^{5} + \frac{27203125503971175436358820643}{138624775427094249276782995174} a^{4} + \frac{45388641342612370703329646627}{138624775427094249276782995174} a^{3} + \frac{5285094894393558221318397132}{69312387713547124638391497587} a^{2} - \frac{68651661664309184052117709813}{138624775427094249276782995174} a - \frac{2497773124445228594092989735}{6301126155777011330762863417}$

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

Class group and class number

$C_{3}\times C_{6}$, which has order $18$ (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:  \( -1 \) (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:  \( 163929.43344817942 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3^2$ (as 18T11):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 9 conjugacy class representatives for $S_3^2$
Character table for $S_3^2$

Intermediate fields

\(\Q(\sqrt{-51}) \), 3.1.108.1, 3.1.459.1 x3, 6.0.171915696.7, 6.0.10744731.1, 9.1.167102056512.1 x3

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 sibling: data not computed
Degree 9 sibling: data not computed
Degree 12 sibling: 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 R R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$2.6.4.1$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}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
3Data not computed
$17$17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$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}$