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Properties

Label 18.0.37621770755...0192.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{20}\cdot 3^{9}\cdot 67^{10}$
Root discriminant $38.68$
Ramified primes $2, 3, 67$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_3\times S_3^2$ (as 18T46)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![106677, -139320, -108675, 192240, 22194, -79632, -12096, 14472, 13300, -3960, -2786, 32, 370, 152, -20, -16, -5, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 5*x^16 - 16*x^15 - 20*x^14 + 152*x^13 + 370*x^12 + 32*x^11 - 2786*x^10 - 3960*x^9 + 13300*x^8 + 14472*x^7 - 12096*x^6 - 79632*x^5 + 22194*x^4 + 192240*x^3 - 108675*x^2 - 139320*x + 106677)
 
gp: K = bnfinit(x^18 - 5*x^16 - 16*x^15 - 20*x^14 + 152*x^13 + 370*x^12 + 32*x^11 - 2786*x^10 - 3960*x^9 + 13300*x^8 + 14472*x^7 - 12096*x^6 - 79632*x^5 + 22194*x^4 + 192240*x^3 - 108675*x^2 - 139320*x + 106677, 1)
 

Normalized defining polynomial

\( x^{18} - 5 x^{16} - 16 x^{15} - 20 x^{14} + 152 x^{13} + 370 x^{12} + 32 x^{11} - 2786 x^{10} - 3960 x^{9} + 13300 x^{8} + 14472 x^{7} - 12096 x^{6} - 79632 x^{5} + 22194 x^{4} + 192240 x^{3} - 108675 x^{2} - 139320 x + 106677 \)

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:  \(-37621770755235979566165000192=-\,2^{20}\cdot 3^{9}\cdot 67^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.68$
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:  $2, 3, 67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{6} a^{6} + \frac{1}{6} a^{4} + \frac{1}{6} a^{2} - \frac{1}{2}$, $\frac{1}{6} a^{7} + \frac{1}{6} a^{5} + \frac{1}{6} a^{3} - \frac{1}{2} a$, $\frac{1}{18} a^{8} + \frac{1}{18} a^{7} + \frac{1}{18} a^{6} + \frac{7}{18} a^{5} - \frac{5}{18} a^{4} + \frac{1}{18} a^{3} - \frac{1}{6} a^{2} + \frac{1}{6} a$, $\frac{1}{18} a^{9} + \frac{1}{3} a^{5} - \frac{2}{9} a^{3} - \frac{1}{6} a$, $\frac{1}{18} a^{10} + \frac{4}{9} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{18} a^{11} + \frac{4}{9} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{108} a^{12} - \frac{1}{54} a^{10} - \frac{1}{54} a^{9} + \frac{1}{108} a^{8} - \frac{1}{54} a^{7} - \frac{1}{18} a^{6} + \frac{23}{54} a^{5} + \frac{5}{12} a^{4} + \frac{7}{18} a^{3} + \frac{1}{3} a - \frac{1}{4}$, $\frac{1}{324} a^{13} + \frac{1}{81} a^{11} - \frac{1}{162} a^{10} + \frac{1}{324} a^{9} - \frac{1}{162} a^{8} + \frac{1}{27} a^{7} - \frac{2}{81} a^{6} + \frac{37}{108} a^{5} - \frac{1}{27} a^{4} + \frac{2}{9} a^{3} + \frac{5}{18} a^{2} + \frac{1}{12} a - \frac{1}{2}$, $\frac{1}{324} a^{14} + \frac{1}{324} a^{12} - \frac{1}{162} a^{11} + \frac{7}{324} a^{10} + \frac{1}{81} a^{9} - \frac{1}{36} a^{8} - \frac{5}{81} a^{7} + \frac{1}{108} a^{6} + \frac{4}{27} a^{5} - \frac{1}{4} a^{4} - \frac{1}{6} a^{3} - \frac{1}{12} a^{2} + \frac{1}{4}$, $\frac{1}{648} a^{15} - \frac{1}{648} a^{14} - \frac{1}{648} a^{13} - \frac{1}{216} a^{12} + \frac{1}{648} a^{11} - \frac{17}{648} a^{10} - \frac{5}{216} a^{9} + \frac{11}{648} a^{8} - \frac{37}{648} a^{7} + \frac{25}{648} a^{6} + \frac{5}{72} a^{5} - \frac{79}{216} a^{4} - \frac{17}{72} a^{3} - \frac{11}{72} a^{2} - \frac{1}{8} a + \frac{1}{8}$, $\frac{1}{1944} a^{16} - \frac{1}{1944} a^{15} - \frac{1}{1944} a^{14} - \frac{1}{648} a^{13} + \frac{1}{1944} a^{12} - \frac{53}{1944} a^{11} + \frac{7}{648} a^{10} + \frac{47}{1944} a^{9} - \frac{1}{1944} a^{8} - \frac{155}{1944} a^{7} - \frac{5}{72} a^{6} + \frac{125}{648} a^{5} - \frac{101}{216} a^{4} + \frac{61}{216} a^{3} + \frac{7}{24} a^{2} + \frac{3}{8} a$, $\frac{1}{276203944999126021453272} a^{17} + \frac{3270682003473645593}{30689327222125113494808} a^{16} - \frac{88643657044503363989}{276203944999126021453272} a^{15} + \frac{335658694543888464539}{276203944999126021453272} a^{14} + \frac{227703084533723112793}{276203944999126021453272} a^{13} - \frac{434306225711242681969}{276203944999126021453272} a^{12} + \frac{928461136570098012529}{276203944999126021453272} a^{11} + \frac{917738568020931904673}{276203944999126021453272} a^{10} + \frac{541417605082788699895}{276203944999126021453272} a^{9} + \frac{1135156111731333928007}{92067981666375340484424} a^{8} - \frac{19434710485793124872567}{276203944999126021453272} a^{7} + \frac{3989595180082024708739}{92067981666375340484424} a^{6} - \frac{32103263581100446623667}{92067981666375340484424} a^{5} + \frac{10868719446310691114321}{30689327222125113494808} a^{4} - \frac{14627602693297580080217}{30689327222125113494808} a^{3} - \frac{603783882976131988537}{3409925246902790388312} a^{2} - \frac{45842678820024999693}{284160437241899199026} a + \frac{232664234173614755169}{568320874483798398052}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( \frac{13149390871}{101161545473124} a^{17} + \frac{17105954305}{202323090946248} a^{16} - \frac{108972610469}{202323090946248} a^{15} - \frac{170418584869}{67441030315416} a^{14} - \frac{925569130669}{202323090946248} a^{13} + \frac{3251323147535}{202323090946248} a^{12} + \frac{3976104782173}{67441030315416} a^{11} + \frac{11085754317583}{202323090946248} a^{10} - \frac{63407050411535}{202323090946248} a^{9} - \frac{151288649895223}{202323090946248} a^{8} + \frac{69728492846111}{67441030315416} a^{7} + \frac{166325193411949}{67441030315416} a^{6} + \frac{25138580073269}{22480343438472} a^{5} - \frac{208906423498651}{22480343438472} a^{4} - \frac{3450147519897}{832605312536} a^{3} + \frac{41748836071829}{2497815937608} a^{2} + \frac{3757648791533}{2497815937608} a - \frac{3608743361031}{416302656268} \) (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:  \( 306761663.9033038 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3^2$ (as 18T46):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 27 conjugacy class representatives for $C_3\times S_3^2$
Character table for $C_3\times S_3^2$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.804.1, 6.0.1939248.2, 6.0.1939248.3

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

Sibling fields

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$2$2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
2.12.16.3$x^{12} - 30 x^{10} - 5 x^{8} + 19 x^{4} + 30 x^{2} + 1$$6$$2$$16$$C_6\times S_3$$[2]_{3}^{6}$
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
$67$67.3.2.2$x^{3} + 268$$3$$1$$2$$C_3$$[\ ]_{3}$
67.3.0.1$x^{3} - x + 16$$1$$3$$0$$C_3$$[\ ]^{3}$
67.6.5.3$x^{6} - 17152$$6$$1$$5$$C_6$$[\ ]_{6}$
67.6.3.1$x^{6} - 134 x^{4} + 4489 x^{2} - 76995328$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$