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Properties

Label 18.0.86161979862...0944.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{9}\cdot 17^{9}\cdot 37^{14}$
Root discriminant $188.01$
Ramified primes $2, 3, 17, 37$
Class number $15521328$ (GRH)
Class group $[2, 2, 2, 1940166]$ (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![55649538283, -29651248130, 30681517019, -11750894177, 6915161271, -2056738955, 887867263, -212878813, 74356953, -14614200, 4348726, -702801, 183489, -24157, 5575, -570, 111, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 7*x^17 + 111*x^16 - 570*x^15 + 5575*x^14 - 24157*x^13 + 183489*x^12 - 702801*x^11 + 4348726*x^10 - 14614200*x^9 + 74356953*x^8 - 212878813*x^7 + 887867263*x^6 - 2056738955*x^5 + 6915161271*x^4 - 11750894177*x^3 + 30681517019*x^2 - 29651248130*x + 55649538283)
 
gp: K = bnfinit(x^18 - 7*x^17 + 111*x^16 - 570*x^15 + 5575*x^14 - 24157*x^13 + 183489*x^12 - 702801*x^11 + 4348726*x^10 - 14614200*x^9 + 74356953*x^8 - 212878813*x^7 + 887867263*x^6 - 2056738955*x^5 + 6915161271*x^4 - 11750894177*x^3 + 30681517019*x^2 - 29651248130*x + 55649538283, 1)
 

Normalized defining polynomial

\( x^{18} - 7 x^{17} + 111 x^{16} - 570 x^{15} + 5575 x^{14} - 24157 x^{13} + 183489 x^{12} - 702801 x^{11} + 4348726 x^{10} - 14614200 x^{9} + 74356953 x^{8} - 212878813 x^{7} + 887867263 x^{6} - 2056738955 x^{5} + 6915161271 x^{4} - 11750894177 x^{3} + 30681517019 x^{2} - 29651248130 x + 55649538283 \)

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:  \(-86161979862499093345337569390232247250944=-\,2^{12}\cdot 3^{9}\cdot 17^{9}\cdot 37^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $188.01$
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:  $2, 3, 17, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{1860798385273780992178312145190569588022207487388389716922342519} a^{17} - \frac{209817419592596421331401922836019362743053364439625187274048634}{1860798385273780992178312145190569588022207487388389716922342519} a^{16} + \frac{238190158189796000517015567025337838125079310864032547624293976}{1860798385273780992178312145190569588022207487388389716922342519} a^{15} - \frac{688186739326897166610648812174449445628036170351676655713158040}{1860798385273780992178312145190569588022207487388389716922342519} a^{14} + \frac{24622407363826442022309774661111884229638847380837670461398595}{64165461561164861799252142937605847862834740944427231618011811} a^{13} - \frac{4428646874112368839757143424495438205024071629671703968905381}{64165461561164861799252142937605847862834740944427231618011811} a^{12} + \frac{155007129007892488479509496401659642587808546924850437005481192}{1860798385273780992178312145190569588022207487388389716922342519} a^{11} - \frac{375071604695367476102972363853921158305521440355040642533970656}{1860798385273780992178312145190569588022207487388389716922342519} a^{10} - \frac{204566238394311255856682099846420781182537843367041563035877494}{1860798385273780992178312145190569588022207487388389716922342519} a^{9} + \frac{177156603835476772989729739748145910292734181999705115397877666}{1860798385273780992178312145190569588022207487388389716922342519} a^{8} - \frac{616977353120747115886722558833115591316728981914100867036633562}{1860798385273780992178312145190569588022207487388389716922342519} a^{7} - \frac{611251105466285246951216522760101901156976267567769585185994368}{1860798385273780992178312145190569588022207487388389716922342519} a^{6} - \frac{4540035605491840864537706805393066548951217518714269566929039}{1860798385273780992178312145190569588022207487388389716922342519} a^{5} - \frac{368493070498522576176795239341640970568065221020399973698608172}{1860798385273780992178312145190569588022207487388389716922342519} a^{4} - \frac{474036274171068554227701661045757909526948051405852251321445541}{1860798385273780992178312145190569588022207487388389716922342519} a^{3} - \frac{522568902629058503660208016630200484634436747008403966500045817}{1860798385273780992178312145190569588022207487388389716922342519} a^{2} + \frac{892786471161841305634295929215521643563627020632858194270024149}{1860798385273780992178312145190569588022207487388389716922342519} a + \frac{299491241882148549995266452116051100518739240363710836009809983}{1860798385273780992178312145190569588022207487388389716922342519}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{1940166}$, which has order $15521328$ (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

\(\Q(\sqrt{-51}) \), 3.3.1369.1, 3.3.148.1, 6.0.248609330811.4, 6.0.2905587504.4, 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 R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
2Data not computed
$3$3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$17$17.6.3.2$x^{6} - 289 x^{2} + 14739$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
17.12.6.1$x^{12} + 117912 x^{6} - 1419857 x^{2} + 3475809936$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$37$37.6.4.1$x^{6} + 518 x^{3} + 171125$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
37.12.10.1$x^{12} + 1998 x^{6} + 21390625$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$