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Properties

Label 18.0.87039873532...3504.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 11^{9}\cdot 37^{14}$
Root discriminant $87.32$
Ramified primes $2, 11, 37$
Class number $20736$ (GRH)
Class group $[2, 6, 24, 72]$ (GRH)
Galois group $S_3 \times C_6$ (as 18T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2999263, -5296030, 6680809, -5640557, 4043761, -2351635, 1224513, -535373, 216453, -81340, 32356, -11321, 3019, -337, -25, -10, 21, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 7*x^17 + 21*x^16 - 10*x^15 - 25*x^14 - 337*x^13 + 3019*x^12 - 11321*x^11 + 32356*x^10 - 81340*x^9 + 216453*x^8 - 535373*x^7 + 1224513*x^6 - 2351635*x^5 + 4043761*x^4 - 5640557*x^3 + 6680809*x^2 - 5296030*x + 2999263)
 
gp: K = bnfinit(x^18 - 7*x^17 + 21*x^16 - 10*x^15 - 25*x^14 - 337*x^13 + 3019*x^12 - 11321*x^11 + 32356*x^10 - 81340*x^9 + 216453*x^8 - 535373*x^7 + 1224513*x^6 - 2351635*x^5 + 4043761*x^4 - 5640557*x^3 + 6680809*x^2 - 5296030*x + 2999263, 1)
 

Normalized defining polynomial

\( x^{18} - 7 x^{17} + 21 x^{16} - 10 x^{15} - 25 x^{14} - 337 x^{13} + 3019 x^{12} - 11321 x^{11} + 32356 x^{10} - 81340 x^{9} + 216453 x^{8} - 535373 x^{7} + 1224513 x^{6} - 2351635 x^{5} + 4043761 x^{4} - 5640557 x^{3} + 6680809 x^{2} - 5296030 x + 2999263 \)

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:  \(-87039873532075611618416695574933504=-\,2^{12}\cdot 11^{9}\cdot 37^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $87.32$
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:  $2, 11, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{6} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{7} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{87} a^{16} + \frac{2}{29} a^{14} + \frac{1}{29} a^{13} - \frac{7}{87} a^{12} + \frac{4}{87} a^{11} - \frac{3}{29} a^{10} - \frac{6}{29} a^{9} + \frac{1}{87} a^{8} - \frac{37}{87} a^{7} - \frac{16}{87} a^{6} + \frac{4}{87} a^{5} - \frac{5}{87} a^{4} - \frac{4}{87} a^{3} + \frac{19}{87} a^{2} - \frac{38}{87} a + \frac{7}{29}$, $\frac{1}{6235939549411415643432947475335681915097} a^{17} - \frac{8219229827781110904131208566619465097}{6235939549411415643432947475335681915097} a^{16} - \frac{56642825231309869726884884151664821500}{6235939549411415643432947475335681915097} a^{15} - \frac{340471757412363631610610811792331733403}{2078646516470471881144315825111893971699} a^{14} - \frac{40508340821479245628863263754363739862}{6235939549411415643432947475335681915097} a^{13} - \frac{493607256147469382680870383924399669571}{6235939549411415643432947475335681915097} a^{12} + \frac{2374359438449445171553243183587358567637}{6235939549411415643432947475335681915097} a^{11} + \frac{14321660611956086941303264631714720057}{6235939549411415643432947475335681915097} a^{10} + \frac{814544696858503363040532016591633029070}{6235939549411415643432947475335681915097} a^{9} + \frac{2409546649705596532253116819308329617705}{6235939549411415643432947475335681915097} a^{8} - \frac{303027539348394160677723179519411369342}{2078646516470471881144315825111893971699} a^{7} - \frac{2431962411928398636533647910100974713934}{6235939549411415643432947475335681915097} a^{6} - \frac{2084052038079738688758058010334127318306}{6235939549411415643432947475335681915097} a^{5} + \frac{908539509363021724998002520334958542240}{6235939549411415643432947475335681915097} a^{4} + \frac{1071891372928423395837466240522055048902}{6235939549411415643432947475335681915097} a^{3} + \frac{1096196369010306709646168307510594506921}{6235939549411415643432947475335681915097} a^{2} - \frac{3000937956219493447343600364774125202655}{6235939549411415643432947475335681915097} a - \frac{2019106963083356380326238662949554843752}{6235939549411415643432947475335681915097}$

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

Class group and class number

$C_{2}\times C_{6}\times C_{24}\times C_{72}$, which has order $20736$ (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:  \( 615797.1340659427 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6\times S_3$ (as 18T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 18 conjugacy class representatives for $S_3 \times C_6$
Character table for $S_3 \times C_6$

Intermediate fields

\(\Q(\sqrt{-11}) \), 3.3.1369.1, 3.3.148.1, 6.0.29154224.2, 6.0.2494508291.2, 9.9.6075640136512.1

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 12 sibling: data not computed
Degree 18 sibling: 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 ${\href{/LocalNumberField/3.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
2Data not computed
$11$11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$37$37.3.2.1$x^{3} - 37$$3$$1$$2$$C_3$$[\ ]_{3}$
37.3.2.1$x^{3} - 37$$3$$1$$2$$C_3$$[\ ]_{3}$
37.6.5.1$x^{6} - 37$$6$$1$$5$$C_6$$[\ ]_{6}$
37.6.5.1$x^{6} - 37$$6$$1$$5$$C_6$$[\ ]_{6}$