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Properties

Label 18.0.50809695180...3536.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{21}\cdot 17^{9}$
Root discriminant $23.58$
Ramified primes $2, 3, 17$
Class number $2$ (GRH)
Class group $[2]$ (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![348, -144, 1020, 384, 660, 1050, 115, 294, 69, -414, -36, -66, -95, 114, 96, -6, -15, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 15*x^16 - 6*x^15 + 96*x^14 + 114*x^13 - 95*x^12 - 66*x^11 - 36*x^10 - 414*x^9 + 69*x^8 + 294*x^7 + 115*x^6 + 1050*x^5 + 660*x^4 + 384*x^3 + 1020*x^2 - 144*x + 348)
 
gp: K = bnfinit(x^18 - 15*x^16 - 6*x^15 + 96*x^14 + 114*x^13 - 95*x^12 - 66*x^11 - 36*x^10 - 414*x^9 + 69*x^8 + 294*x^7 + 115*x^6 + 1050*x^5 + 660*x^4 + 384*x^3 + 1020*x^2 - 144*x + 348, 1)
 

Normalized defining polynomial

\( x^{18} - 15 x^{16} - 6 x^{15} + 96 x^{14} + 114 x^{13} - 95 x^{12} - 66 x^{11} - 36 x^{10} - 414 x^{9} + 69 x^{8} + 294 x^{7} + 115 x^{6} + 1050 x^{5} + 660 x^{4} + 384 x^{3} + 1020 x^{2} - 144 x + 348 \)

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:  \(-5080969518089676267073536=-\,2^{12}\cdot 3^{21}\cdot 17^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.58$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{12} a^{14} + \frac{1}{12} a^{13} + \frac{1}{12} a^{12} + \frac{1}{6} a^{11} - \frac{1}{12} a^{10} - \frac{1}{12} a^{9} + \frac{1}{12} a^{8} + \frac{1}{3} a^{7} + \frac{1}{12} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a$, $\frac{1}{12} a^{15} + \frac{1}{12} a^{12} - \frac{1}{4} a^{11} + \frac{1}{6} a^{9} + \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{3} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{180} a^{16} + \frac{1}{90} a^{15} + \frac{1}{60} a^{14} + \frac{7}{45} a^{13} + \frac{7}{90} a^{12} - \frac{1}{30} a^{11} - \frac{1}{180} a^{10} + \frac{7}{45} a^{9} + \frac{1}{30} a^{8} + \frac{4}{9} a^{7} - \frac{47}{180} a^{6} + \frac{1}{3} a^{5} + \frac{3}{20} a^{4} - \frac{1}{5} a^{3} + \frac{11}{30} a^{2} - \frac{13}{30} a - \frac{1}{15}$, $\frac{1}{14134497064882599914100} a^{17} - \frac{29603595743989341349}{14134497064882599914100} a^{16} - \frac{19434570652918621673}{785249836937922217450} a^{15} + \frac{23290760780680286444}{706724853244129995705} a^{14} + \frac{1191993547913533124351}{14134497064882599914100} a^{13} - \frac{36804705603829061819}{157049967387584443490} a^{12} - \frac{301142050046897102983}{1413449706488259991410} a^{11} + \frac{120616653213432238126}{3533624266220649978525} a^{10} + \frac{2777934272168671727}{1570499673875844434900} a^{9} - \frac{619041084876080329624}{3533624266220649978525} a^{8} - \frac{2964334374745158667201}{7067248532441299957050} a^{7} + \frac{16453555987479826297}{392624918468961108725} a^{6} + \frac{34145598005786824087}{392624918468961108725} a^{5} + \frac{59051708540625442023}{1570499673875844434900} a^{4} + \frac{1058220739614151482407}{2355749510813766652350} a^{3} + \frac{382461858905026887221}{2355749510813766652350} a^{2} + \frac{722725693088514859291}{2355749510813766652350} a + \frac{158105720876041130282}{392624918468961108725}$

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:  $8$
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:  \( 219359.51828424638 \) (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, 3.1.204.1 x3, 6.0.171915696.7, 6.0.2122416.1, 9.1.18566895168.1 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$$[\ ]_{3}^{2}$
$3$3.6.7.2$x^{6} + 3 x^{2} + 6$$6$$1$$7$$D_{6}$$[3/2]_{2}^{2}$
3.6.7.2$x^{6} + 3 x^{2} + 6$$6$$1$$7$$D_{6}$$[3/2]_{2}^{2}$
3.6.7.2$x^{6} + 3 x^{2} + 6$$6$$1$$7$$D_{6}$$[3/2]_{2}^{2}$
$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}$