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Properties

Label 18.14.1198960776...0000.1
Degree $18$
Signature $[14, 2]$
Discriminant $2^{12}\cdot 3^{24}\cdot 5^{9}\cdot 79^{2}\cdot 149\cdot 7554091^{2}$
Root discriminant $191.49$
Ramified primes $2, 3, 5, 79, 149, 7554091$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group 18T874

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1914919780, 5068342260, 5719657296, 3698753736, 1510844220, 261301800, -160347317, -125806305, -19549308, 10055456, 3594435, -273006, -226799, -1860, 7479, 222, -132, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 132*x^16 + 222*x^15 + 7479*x^14 - 1860*x^13 - 226799*x^12 - 273006*x^11 + 3594435*x^10 + 10055456*x^9 - 19549308*x^8 - 125806305*x^7 - 160347317*x^6 + 261301800*x^5 + 1510844220*x^4 + 3698753736*x^3 + 5719657296*x^2 + 5068342260*x + 1914919780)
 
gp: K = bnfinit(x^18 - 3*x^17 - 132*x^16 + 222*x^15 + 7479*x^14 - 1860*x^13 - 226799*x^12 - 273006*x^11 + 3594435*x^10 + 10055456*x^9 - 19549308*x^8 - 125806305*x^7 - 160347317*x^6 + 261301800*x^5 + 1510844220*x^4 + 3698753736*x^3 + 5719657296*x^2 + 5068342260*x + 1914919780, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 132 x^{16} + 222 x^{15} + 7479 x^{14} - 1860 x^{13} - 226799 x^{12} - 273006 x^{11} + 3594435 x^{10} + 10055456 x^{9} - 19549308 x^{8} - 125806305 x^{7} - 160347317 x^{6} + 261301800 x^{5} + 1510844220 x^{4} + 3698753736 x^{3} + 5719657296 x^{2} + 5068342260 x + 1914919780 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[14, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(119896077691376921345726910546792000000000=2^{12}\cdot 3^{24}\cdot 5^{9}\cdot 79^{2}\cdot 149\cdot 7554091^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $191.49$
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:  $2, 3, 5, 79, 149, 7554091$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{204} a^{15} - \frac{1}{4} a^{14} - \frac{6}{17} a^{13} + \frac{10}{51} a^{12} - \frac{29}{68} a^{11} + \frac{2}{17} a^{10} - \frac{67}{204} a^{9} + \frac{1}{17} a^{8} + \frac{1}{4} a^{7} - \frac{19}{102} a^{6} - \frac{1}{4} a^{4} + \frac{83}{204} a^{3} - \frac{6}{17} a^{2} - \frac{1}{34} a - \frac{1}{6}$, $\frac{1}{10812} a^{16} - \frac{1}{1802} a^{15} - \frac{789}{3604} a^{14} - \frac{1004}{2703} a^{13} + \frac{1047}{3604} a^{12} + \frac{23}{68} a^{11} - \frac{5107}{10812} a^{10} + \frac{495}{3604} a^{9} + \frac{197}{3604} a^{8} + \frac{4909}{10812} a^{7} - \frac{387}{1802} a^{6} - \frac{73}{212} a^{5} - \frac{655}{2703} a^{4} + \frac{1017}{3604} a^{3} + \frac{139}{1802} a^{2} + \frac{740}{2703} a + \frac{3}{106}$, $\frac{1}{2663018225621043731401336561496940} a^{17} + \frac{407070492543117405515567483}{10443208727925661691769947299988} a^{16} - \frac{1008666266838201967088731121379}{887672741873681243800445520498980} a^{15} - \frac{1013973275423738361366965650357829}{2663018225621043731401336561496940} a^{14} - \frac{54581055039535065710092145865661}{887672741873681243800445520498980} a^{13} + \frac{34117669328050594825317941239613}{221918185468420310950111380124745} a^{12} + \frac{295468688226650456532451167831026}{665754556405260932850334140374235} a^{11} - \frac{76033871591590974250276322909097}{221918185468420310950111380124745} a^{10} - \frac{216944105959053856270215550359097}{443836370936840621900222760249490} a^{9} - \frac{73909943847069522571931177477621}{266301822562104373140133656149694} a^{8} + \frac{22055019446588590680613005767647}{52216043639628308458849736499940} a^{7} + \frac{310701627279748466277196495402097}{887672741873681243800445520498980} a^{6} + \frac{791869612184621848289103613164371}{2663018225621043731401336561496940} a^{5} + \frac{52957642160944536069278044455221}{887672741873681243800445520498980} a^{4} - \frac{156564065703570489410738392792227}{887672741873681243800445520498980} a^{3} + \frac{398302668833761386447953383206439}{1331509112810521865700668280748470} a^{2} - \frac{43508101804261259984602961545455}{88767274187368124380044552049898} a + \frac{1106240114079176660403039511799}{5221604363962830845884973649994}$

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:  $15$
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:  \( 236936014565000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T874:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 279936
The 174 conjugacy class representatives for t18n874 are not computed
Character table for t18n874 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 3.3.1620.1 x3, 6.6.13122000.1

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

Sibling fields

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 R ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.9.0.1}{9} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.9.0.1}{9} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/41.9.0.1}{9} }{,}\,{\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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$$[\ ]_{3}^{2}$
2.12.8.1$x^{12} - 6 x^{9} + 12 x^{6} - 8 x^{3} + 16$$3$$4$$8$$C_3 : C_4$$[\ ]_{3}^{4}$
$3$3.6.8.5$x^{6} + 9 x^{2} + 9$$3$$2$$8$$S_3$$[2]^{2}$
3.12.16.30$x^{12} + 93 x^{11} + 351 x^{10} + 3 x^{9} + 126 x^{8} - 297 x^{7} + 171 x^{6} + 243 x^{5} - 324 x^{4} - 54 x^{3} + 162 x^{2} - 243 x + 324$$3$$4$$16$$C_3 : C_4$$[2]^{4}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$79$$\Q_{79}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{79}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{79}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{79}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{79}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{79}$$x + 2$$1$$1$$0$Trivial$[\ ]$
79.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
79.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
79.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
79.3.0.1$x^{3} - x + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
79.3.2.3$x^{3} - 316$$3$$1$$2$$C_3$$[\ ]_{3}$
$149$$\Q_{149}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{149}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{149}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{149}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{149}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{149}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{149}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{149}$$x + 2$$1$$1$$0$Trivial$[\ ]$
149.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
149.2.1.1$x^{2} - 149$$2$$1$$1$$C_2$$[\ ]_{2}$
149.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
149.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
149.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7554091Data not computed